THE SINC-GALERKIN METHOD FOR PROBLEMS
IN OCEANOGRAPHY
by
Sanoe Koonprasert
A dissertation submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Mathematics
MONTANA STATE UNIVERSITY
Bozeman, Montana
April 2003
c
°COPYRIGHT
by
Sanoe Koonprasert
2003
All Rights Reserved
ii
APPROVAL
of a dissertation submitted by
Sanoe Koonprasert
This dissertation has been read by each member of the dissertation committee and
has been found to be satisfactory regarding content, English usage, format, citations,
bibliographic style, and consistency, and is ready for submission to the College of
Graduate Studies.
Dr. Kenneth L. Bowers
(Signature)
Date
Approved for the Department of Mathematical Sciences
Dr. Kenneth L. Bowers
(Signature)
Date
Approved for the College of Graduate Studies
Dr. Bruce Mcleod
(Signature)
Date
iii
STATEMENT OF PERMISSION TO USE
In presenting this dissertation in partial fulfillment of the requirements for a
doctoral degree at Montana State University, I agree that the Library shall make it
available to borrowers under rules of the Library. I further agree that copying of
this dissertation is allowable only for scholarly purposes, consistent with “fair use” as
prescribed in the U. S. Copyright Law. Requests for extensive copying or reproduction
of this dissertation should be referred to Bell & Howell Information and Learning,
300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted “the
exclusive right to reproduce and distribute my dissertation in and from microform
along with the non-exclusive right to reproduce and distribute my abstract in any
format in whole or in part.”
Signature
Date
iv
ACKNOWLEDGEMENTS
I would like to thank my sponsor, King Mongkut Institute Technology North
Bangkok, and my family for their complete support throughout my graduate school
years. I also thank James Jacklitch for his LaTeX guidance.
I would like to thank Dr. John Lund, Dr. Jack Dockery, Dr. Mark Pernarowski,
and Dr. Isaac Klapper for their time, suggestions, and comments as my graduate
committee members. I also thank Dr. Don Winter for sharing his extensive knowledge
of oceanography. Lastly, I would like to especially thank my advisor, Dr. Kenneth
Bowers, for his guidance and patience, without which, this manuscript would not
have been possible.
v
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. SINC FUNCTION FUNDAMENTALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
The Sinc Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sinc Basis Functions for the Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . .
Sinc Interpolation and Quadrature Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The General Sinc-Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Matrix Representation of the Derivatives of Sinc Basis Functions at Nodal
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Matrices and Kronecker Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
9
15
19
21
25
3. WIND-DRIVEN CURRENTS IN A SEA WITH A DEPTHDEPENDENT EDDY VISCOSITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sinc-Galerkin Solution of the Complex Velocity Formulation . . . . . . . . . . . .
Numerical Testing: Complex Velocity Formulation . . . . . . . . . . . . . . . . . . . . .
Numerical Testing: Variable Eddy Viscosity in Complex Velocity Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sinc-Galerkin Solution of the Coupled Differential Equation System . . . . . .
Numerical Testing: Coupled System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical Testing: The variable Eddy Viscosity in Coupled System Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. WIND-DRIVEN CURRENTS IN A SEA WITH A TIME-DEPENDENT
EDDY VISCOSITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
35
44
50
56
61
63
69
Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Solving the Problem with Time-Independent Boundary Conditions . . . . . . . 79
Parameter Selections for the Fully Sinc-Galerkin Method . . . . . . . . . . . . . . . 96
Numerical Examples for Time-Independent Boundary Conditions . . . . . . . . 98
Solving the Problem with Time-Dependent Boundary Conditions . . . . . . . . 114
Numerical Examples for Time-Dependent Boundary Conditions . . . . . . . . . 125
5. THE SPIN-UP AND EPISODIC WIND STRESS PROBLEMS . . . . . . . . . 170
The Spin-Up Wind Stress Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
vi
Numerical Steady-State Problem for a No-Slip Bottom Condition . . . . . . . . 178
Numerical Steady-State Problem for a Linear Stress Bottom Condition . . . 185
The Episodic Wind Stress Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
vii
LIST OF TABLES
Table
Page
1. Parameter values corresponding to a range of wind speeds for
oceanography problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2. Errors for Example 1 (constant eddy viscosity) on the sinc grid S
with the linear stress bottom condition for σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3. Errors for Example 2 on the sinc grid S with the zero-velocity bottom
condition (no-slip) for σ = 0, χ = 45o , κ = 5, D0 = 100 m, DE =
20 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4. Errors for Example 5 (constant eddy viscosity) on the sinc grid S
with the linear stress bottom condition for σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5. Errors for Example 6 (constant eddy viscosity) on the sinc grid S
with the zero-velocity bottom condition for σ = 0, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
6. Comparison between the approximate solutions for the complex velocity system and the coupled system by using the same sinc grid
size for Examples 1 and 5 for the case of constant eddy viscosity with linear stress bottom boundary (σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
7. Comparison between the approximate solutions for the complex velocity system and the coupled system on the same sinc grid for
Examples 4 and 8 for the decreasing eddy viscosity A∗v (z ∗ ) =
.02(1 + .12z ∗ (1 − .01z ∗ )) with linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . .
68
8. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choice of Nt for Example 9 with Av (t) ≡ 1 and the
parameter σ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
viii
9. Errors on the sinc grid S and the uniform grid U for the choice
Mz = Nz = Mt = 2Nt for Example 9 with Av (t) ≡ 1 and the
parameter σ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt in Example 10 with Av (t) ≡ 1 and the
parameter σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt of Example 10 with Av (t) ≡ 1 and the
parameter σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
12. Errors on the sinc grid S and the uniform grid U for the choices
t+1
and
Mz = Nz = Mt = 2Nt for Example 11 with Av (t) =
t+2
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
13. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
t+1
and
and various choices of Nt for Example 11 with Av (t) =
t+2
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
14. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 12 with Av (t) = 2 − e−2t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
15. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt for Example 12 with Av (t) = 2 − e−2t and
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
16. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 13 with Av (t) = 2 − e−2t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
17. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt for Example 13 with Av (t) = 2 − e−2t and
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
18. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 14 with Av (t) = 1 + te1−t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
ix
19. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt for Example 14 with Av (t) = 1 + te1−t and
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
20. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 15 with Av (t) = 1 + te1−t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
21. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt for Example 15 with Av (t) = 1 + te1−t and
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
22. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 16 with Av (t) = 1 + te1−t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
23. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 16 with Av (t) = 1 + te1−t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
24. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 17 with Av (t) = 1 + te1−t
and σ = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
25. Errors on the sinc grid S and the uniform grid U for Mz = Nz =
Mt = 2Nt for Example 17 with Av (t) = 1 + te1−t . . . . . . . . . . . . . . . . 154
26. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 18 with Av (t) = 1 + te1−t
and σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
27. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt for Example 18 with Av (t) = 1 + te1−t and
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
28. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt
and various choices of Nt for Example 19 with Av (t) = 1 + te1−t
and σ = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
29. Errors on the sinc grid S and the uniform grid U for the choices
Mz = Nz = Mt = 2Nt for Example 19 with Av (t) = 1 + te1−t and
σ = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
x
30. The convergence results on the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 20 with Av (t) = 4 − 3e−t and
σ = 0, κ = 3.14, fc = 4.95, and D0 = 60 m . . . . . . . . . . . . . . . . . . . . . 175
31. The errors of Example 20 (at steady-state) on the uniform grid Uz for
the choices Mz = Nz = Mt = 2Nt = 4, 8, 16, 32 and tT = 110
with a no-slip bottom condition (σ = 0), κ = 3.14, fc = 4.95,
and D0 = 60 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
32. The convergence results on the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 21 with Av (t) = 4 − 3e−t and
σ = 0.16, κ = 3.14, fc = 4.95, and D0 = 60 m . . . . . . . . . . . . . . . . . . 182
33. The errors of Example 21 (at steady-state) on the uniform grid Uz for
the choices Mz = Nz = Mt = 2Nt = 4, 8, 16, 32 and tT = 110
with the linear stress bottom condition (σ = 0.16), κ = 3.14,
fc = 4.95, and D0 = 60 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
34. The convergence results on the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 22 with Av (t) = 1 + 3te1−t and
σ = 0, κ = 3.14, fc = 3.2, and D0 = 60 m . . . . . . . . . . . . . . . . . . . . . . 190
35. The convergence results on the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 23 with Av (t) = 1 + 3te1−t and
σ = 0.16, κ = 3.14, fc = 3.2, and D0 = 60 m . . . . . . . . . . . . . . . . . . . 194
xi
LIST OF FIGURES
Figure
Page
1. The basis functions S(k, h)(x) for k = −1, 0, 1, with h = π/4 . . . . . . .
8
2. Central sinc basis function S(0, h)(x) for h = π/2, π/4, π/8 . . . . . . . . .
8
3. The relationship between the eye-shaped domain, DE , and the infinite
strip, DS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4. Three adjacent members S(k, h) ◦ φ(x) when k = −1, 0, 1 and h = π8
of the mapped sinc basis on the interval (0, 1) . . . . . . . . . . . . . . . . . .
5. The relationship between the grid points xk ∈ (0, 1) and the grid
points kh ∈ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
S(k, h) ◦ φ(x)
when k = −1, 0, 1 and h = π8
φ0 (x)
of the modified sinc basis on the interval (0, 1) . . . . . . . . . . . . . . . . . .
11
12
6. Three adjacent members
12
7. The relationship between the wedge-shaped domain, DW , and the
infinite strip, DS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
8. Three adjacent members S(k, h) ◦ Υ(t) when k = −1, 0, 1 and h = π8
of the mapped sinc basis on (0, ∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
9. The relationship between the grid points tk ∈ (0, ∞) and the grid
points kh ∈ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
10. The general physical model of the depth-dependent eddy viscosity
oceanography problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
11. Sinc-Galerkin Ekman spiral projections for Example 1 with increasing
N for constant eddy viscosity with linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . .
48
12. Sinc-Galerkin Ekman spiral projections for Example 2 with increasing
N for constant eddy viscosity with no-slip bottom boundary (σ =
0, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . . . . . . . .
50
xii
13. Eddy viscosity functions A∗v (z ∗ ) = 0.02(1 − .0075z ∗ )2 (m2 /s) and
A∗v (z ∗ ) ≡ 0.02 (m2 /s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
14. Sinc-Galerkin Ekman spiral projections for Example 3 with both
constant and decreasing eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE =
20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
15. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 3 with constant and decreasing eddy viscosity functions
and linear stress bottom boundary (σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
16. Sinc-Galerkin Ekman spiral projections for Example 3 with increasing
N for the case of the decreasing eddy viscosity function with
linear stress bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 =
100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
17. Eddy viscosity functions A∗v (z ∗ ) = 0.02[1 + (.12)z ∗ (1 − (.01)z ∗ )] and
A∗v (z ∗ ) ≡ 0.02 (m2 /s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
18. Sinc-Galerkin Ekman spiral projections for Example 4 with both
constant and quadratic eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE =
20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
19. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 4 with constant and quadratic eddy viscosity functions
and linear stress bottom boundary (σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
20. Sinc-Galerkin Ekman spiral projections for Example 4 with increasing N for the case of the quadratic eddy viscosity function with
linear stress bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 =
100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
21. Sinc-Galerkin Ekman spiral projections for Example 5 with increasing N for the case of constant eddy viscosity, A∗v (z ∗ ) ≡ 1, with
linear stress bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 =
100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
xiii
22. Sinc-Galerkin Ekman spiral projections for Example 6 with increasing N for the case of constant eddy viscosity, A∗v (z ∗ ) ≡ 1,
with no-slip bottom boundary (σ = 0, χ = 45o , κ = 5, D0 =
100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
23. Sinc-Galerkin Ekman spiral projections for Example 7 with both
constant and decreasing eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE =
20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
24. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 7 with constant and decreasing eddy viscosity functions
and linear stress bottom boundary (σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
25. Sinc-Galerkin Ekman spiral projections for Example 7 with increasing N for the case of the decreasing eddy viscosity function and
linear stress bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 =
100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
26. Sinc-Galerkin Ekman spiral projections for Example 8 with both
constant and quadratic eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE =
20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
27. Sinc-Galerkin northward and eastward calculated velocity profiles
for Example 8 with constant and quadratic eddy viscosity functions and linear stress bottom boundary(σ = 0.1, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
28. Sinc-Galerkin Ekman spiral projections for Example 8 with increasing N for the case of the quadratic eddy viscosity function and
linear stress bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 =
100 m, DE = 20 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
29. The general physical model of time-dependent eddy viscosity
oceanography problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
30. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 9 where Av (t) ≡ 1 and σ = 0 with Mz = Nz = Mt =
16 , Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xiv
31. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 9 where Av (t) ≡ 1 and σ = 0 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
32. The approximate solution wa (z, t) when z = 0, .16, .50, and 1 on
the sinc grid S t for Example 9 with Mz = Nz = Mt = 16, and
Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
33. The approximate solution of wa (z, t) at z = 0, .20, .50, and 1 on
the uniform grid U t for Example 9 with Mz = Nz = Mt = 16,
and Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
34. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 10 where Av (t) ≡ 1 and σ = 1 with Mz = Nz = Mt =
16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
35. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 10 where Av (t) ≡ 1 and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
36. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 10 with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . 108
37. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 10 with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . 109
38. The graph of the approximate solution wa (z, t) on the sinc grid S
t+1
for Example 11 where Av (t) =
and σ = 1 with Mz = Nz =
t+2
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
39. The graph of the approximate solution wa (z, t) on the uniform grid
t+1
U for Example 11 where Av (t) =
and σ = 1 with Mz =
t+2
Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
40. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
t+1
grid S t for Example 11 where Av (t) =
and σ = 1 with
t+2
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
41. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
t+1
grid U t for Example 11 where Av (t) =
and σ = 1 with
t+2
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
xv
42. The graph of the time-dependent increasing eddy viscosity Av (t) =
2 − e−2t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
43. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 12 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
44. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 12 where Av (t) = 2 − e−2t and σ = 1 with Mz =
Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
45. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 12 where Av (t) = 2 − e−2t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
46. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 12 where Av (t) = 2 − e−2t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
47. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 13 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
48. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 13 where Av (t) = 2 − e−2t and σ = 1 with Mz =
Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
49. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 13 where Av (t) = 2 − e−2t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
50. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 13 where Av (t) = 2 − e−2t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
51. The graph of the time-dependent eddy viscosity Av (t) = 1 + te1−t
with a zero steady-state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
52. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 14 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
xvi
53. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 14 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
54. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 14 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
55. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 14 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
56. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 15 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
57. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 15 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
58. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 15 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
59. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 15 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
60. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 16 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
61. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 16 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
62. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 16 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
63. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 16 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xvii
64. The graph of the approximate solution (real part) ua (z, t) on the sinc
grid S for Example 17 with Av (t) = 1 + te1−t with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
65. The graph of the approximate solution (imaginary part) va (z, t) on
the sinc grid S for Example 17 with Av (t) = 1 + te1−t with Mz =
Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
66. The graph of the approximate solution (real part) ua (z, t) on the
uniform grid U for Example 17 with Av (t) = 1 + te1−t with Mz =
Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
67. The graph of the approximate solution (imaginary part) va (z, t) on
the uniform grid U for Example 17 with Av (t) = 1 + te1−t with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
68. The approximate solution (real part) ua (z, t) at z = 0, .16, .50, 1
on the sinc grid S t for Example 17 where Av (t) = 1 + te1−t with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
69. The approximate solution (imaginary part) va (z, t) at z =
0, .16, .50, 1 on the sinc grid S t for Example 17 where Av (t) =
1 + te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . 156
70. The approximate solution (real part) ua (z, t) at z = 0, .20, .50, 1
on the uniform grid U t for Example 17 where Av (t) = 1 + te1−t
with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
71. The approximate solution (imaginary part) va (z, t) at z =
0, .20, .50, 1 on the uniform grid U t for Example 17 where
Av (t) = 1 + te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . 157
72. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 18 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
73. The graph of the approximate solution wa (z, t) on the uniform grid
U for Example 18 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
74. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc
grid S t for Example 18 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
xviii
75. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 18 where Av (t) = 1 + te1−t and σ = 1 with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
76. The graph of the approximate solution (real part) ua (z, t) on the sinc
grid S for Example 19 with Av (t) = 1 + 3te1−t with Mz = Nz =
Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
77. The graph of the approximate solution (imaginary part) va (z, t) on
the sinc grid S for Example 19 where Av (t) = 1 + 3te1−t with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
78. The graph of the approximate solution (real part) ua (z, t) on the
uniform grid U for Example 19 with Av (t) = 1 + 3te1−t with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
79. The graph of the approximates solution (imaginary part) va (z, t) on
the uniform grid U for Example 19 where Av (t) = 1 + te1−t with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
80. The approximate solution (real part) ua (z, t) at z = 0, .16, .50, 1
on the sinc grid S t for Example 19 where Av (t) = 1 + 3te1−t with
Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
81. The approximate solution (imaginary part) va (z, t) at z =
0, .16, .50, 1 on the sinc grid S t for Example 19 where Av (t) =
1 + te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . 168
82. The approximate solution (real part) ua (z, t) at z = 0, .20, .50, 1
on the uniform grid U t for Example 19 where Av (t) = 1 + 3te1−t
with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
83. The approximate solution (imaginary part) va (z, t) at z =
0, .20, .50, 1 on the uniform grid U t for Example 19 where
Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . 169
∗
−
³ ∗
t /T0
´
84. The graph of the spin-up wind stress ψ(t ) = 1 − e
where
T0 = 12.5 × 3600 = .45 × 105 s is the fundamental time scale of
12.5 hrs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
85. The graph of the nondimensional spin-up wind stress ψ(t) = 1 − e−t
where T0 = 12.5 hrs. is the fundamental time scale, so t0 = 1 . . . . . 174
xix
86. The graph of the approximate northward current component
Ua∗ (z ∗ , t∗ ) (m/s) on the uniform grid U for Example 20 where
Av (t) = 4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . 175
87. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 20 where Av (t) =
4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . 176
88. The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 20
where Av (t) = 4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . 177
89. The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 20
where Av (t) = 4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . 177
90. Fully Sinc-Galerkin Ekman spiral projections for Example 20 on the
uniform grid Uz0 with increasing N ≡ Mz = Nz = Mt = 2Nt for
the steady-state with no-slip bottom boundary condition σ = 0
with χ = π/4, κ = 3.14, D0 = 60 m, DE = 19 m . . . . . . . . . . . . . . . 180
91. The graph of the approximate northward current component
Ua∗ (z ∗ , t∗ ) (m/s) on the uniform grid U for Example 21 where
Av (t) = 4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . 183
92. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 21 where Av (t) =
4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . . . 183
93. The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 21
where Av (t) = 4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . 184
94. The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 21
where Av (t) = 4 − 3e−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . 184
95. Fully Sinc-Galerkin Ekman spiral projections for Example 21 (for the
steady-state) on the uniform grid Uz0 with increasing N = Mz =
Nz = Mt = 2Nt with the linear stress bottom boundary condition
(σ = 0.16), χ = π/4, κ = 3.14, D0 = 60 m, DE = 19 m . . . . . . . . . 187
xx
∗
96. The graph of the episodic wind stress ψ(t∗ ) = (t∗ /T0 )e1−(t /T0 ) where
T0 = 8.1 × 3600 = 2.916 × 104 s is the fundamental time scale of
8.1 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
97. The graph of the nondimensional episodic wind stress ψ(t) = te1−t
where T0 = 8.1 hrs. is the fundamental time scale, so t0 = 1 . . . . . . 189
98. The graph of the approximate northward current component
Ua∗ (z ∗ , t∗ ) (m/s) on the uniform grid U for Example 22 where
Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . 191
99. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 22 where Av (t) =
1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . 191
100.The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 22
where Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . 192
101.The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 22
where Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . 192
102.The graph of the approximate northward current component
Ua∗ (z ∗ , t∗ ) (m/s) on the uniform grid U for Example 23 where
Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . 194
103.The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 23 where Av (t) =
1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . . . . . . . . . . . . . . 195
104.The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 23
where Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . 196
105.The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at
z ∗ = 0, 19, 30, 60 m on the uniform grid U t for Example 23
where Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8 . . . . . . 196
xxi
ABSTRACT
The model for a wind-driven current with depth-dependent eddy viscosity is developed as a complex velocity formulation and a coupled system formulation. The
numerical solutions, calculated by a Sinc-Galerkin method, are compared. A fully
Sinc-Galerkin method in both space and time for a partial differential equation with
time-dependent boundary conditions is developed. This entirely new technique is
applied to the model for a wind-driven current with time-dependent viscosity. Several numerical examples are used to test the performance of the method. This fully
Sinc-Galerkin method is then applied to the spin-up and the episodic wind stress
oceanography problems.
1
CHAPTER 1
INTRODUCTION
The intent of this thesis is to describe a model of wind-driven currents in the
ocean and to develop an efficient and highly accurate numerical scheme to solve the
model. This will lead to advances in the understanding of ocean circulation and
to the introduction of a novel numerical technique in oceanography, the fully SincGalerkin method. This method may well provide an alternate approach to problems
in numerical oceanography.
A thorough development of the model described here can be found in [25]. In
general terms, a strong wind (storm) blows over the ocean and energy is transferred
from the wind to the surface layers of the ocean. Some of this energy is expended and
leads to a small net movement of water in the direction of the wind. This begins wave
propagation or wind-driven currents. The greater the speed of the wind, the greater
the frictional force acting on the sea surface, and the stronger the surface current.
The frictional force acting on the sea surface is called a wind stress that has been
found by experiment to be proportional to the square of the wind speed. The wind
speed depends on time, denoted by t∗ . Thus if ψ(t∗ ) is the frictional force and Ww (t∗ )
is the wind speed then
ψ(t∗ ) = ρair CD (Ww (t∗ ))2
2
where ρair is the air density and the dimensionless parameter CD = 2.5 × 10−3 .
The complete development of what is outlined here is found in Chapter 4. The
effect of the wind stress at the surface is transmitted downward as a result of internal
friction within the ocean layers. The internal friction in a moving fluid between
different layers of fluid is called the eddy viscosity which also depends on time. This
eddy viscosity, A∗v (t∗ ), is described in terms of the frictional force ψ(t∗ ) by
A∗v (t∗ ) = A0 [1 + kw ψ(t∗ )],
where kw is a magnification parameter and A0 is a turbulent eddy viscosity coefficient
near the surface.
The ocean surface is subject to friction due to the wind stress at its upper surface
and to friction with each successive layer. Each layer is acted upon at its upper surface
by friction with the layer above and by friction with the layer below. Because they
are moving, all layers are acted upon by the Coriolis force. Using Newton’s Second
Law of Motion, and considering the balance of force between friction and the Coriolis
force on the layers making up the water column of depth D0 , the horizontal wind
drift current velocity q ∗ is determined by solving the initial-boundary-value problem
µ
¶
∗ ∗ ∗
∂
∂q ∗ (z ∗ , t∗ )
∗ ∗ ∂q (z , t )
= f ẑ ∗ × q ∗ (z ∗ , t∗ ),
− ∗ Av (t )
∗
∗
∂t
∂z
∂z
0 < z ∗ < D0 ,
Here f is the Coriolis parameter.
0 < t∗ .
3
The stress condition at the sea surface, z ∗ = 0, is equal to the tangential surface
time-dependent wind stress
−ρA∗v (t∗ )
∂q ∗ (0, t∗ )
= τw ψ(t∗ )(cos(χ(t∗ ))x̂∗ + sin(χ(t∗ ))ŷ ∗ ),
∂z ∗
0 < t∗ .
Here ρ is the ocean mass density, τw is the wind stress magnitude, and χ(t∗ ) is the
angle of the wind stress. The frictional stress is assumed linearly proportional to the
current at the seabed, z ∗ = D0 ,
−ρA∗v (t∗ )
∂q ∗ (D0 , t∗ )
= kf ρq ∗ (D0 , t∗ ),
∂z ∗
0 < t∗ .
Here kf is the linear slip bottom stress coefficient. Initially the sea is assumed to be
at rest, so
q ∗ (z ∗ , 0) = 0,
0 < z ∗ < D0 .
This model can be simplified by rescaling and nondimensionalizing the parameters, variables, and functions. Also, it can be transformed to a complex velocity
formulation. Then the resulting model is formulated with the operator
Hw(z, t)
≡
µ
¶
fc ∂
∂w(z, t)
∂w(z, t)
− 2
Av (t)
.
∂t
2κ ∂z
∂z
The model becomes
Hw(z, t) − ifc w(z, t)
=
F (z, t), 0 < z < 1, 0 < t
4
where fc = f T0 , T0 is the fundamental time scale, κ ≡ D0 /DE , DE is the Ekman
depth, σ = A0 /(kf D0 ), and
≡
F (z, t)
−κ(1 − z)r(t) − κσg(t).
Here
0
0
r(t) ≡ Φ (t) − ifc Φ(t),
g(t) ≡ Ψ (t) − ifc Ψ(t)
and
Φ(t) ≡
ψ(t) iχ(t)
e
,
Av (t)
Ψ(t) ≡ ψ(t)eiχ(t) .
The surface conditions are
∂w(0, t)
∂z
=
0,
0 < t,
the seabed conditions become
w(1, t) + σAv (t)
∂w(1, t)
∂z
=
0,
0 < t,
and the initial conditions are then
w(z, 0) = 0,
0 < z < 1.
This resulting initial-boundary-value problem is solved via a fully Sinc-Galerkin
method [1], [2], [4], [11], [13], [14], [18], [19], [20], [21], and [22]. It commonly begins
5
with a Galerkin discretization of the spatial and temporal variables. The basis functions in both spaces are the compositions of sinc functions with suitable conformal
maps [12], [23], and [24], that are found in Chapter 2.
The fully Sinc-Galerkin method in space and time has many interesting features, due both to the properties of the basis functions and to the choice of conformal maps and weight functions. Furthermore, the optimal order of convergence,
O(e−k
√
M
), k > 0, using the sinc quadrature rule is maintained in a Sinc-Galerkin
scheme. In addition, the discrete system requires no numerical integrations to compute either the coefficient matrix or the right-hand-side vector. All these features
prove to be advantageous when solving the initial-boundary-value problem.
Chapter 2 begins with a brief review of sinc functions and the associated sinc
quadrature rule which is found in [12] and [24]. The choice of inner product and
weight functions, as well as the Galerkin approximation are discussed. The errors of
approximation in both space and time are presented. The resulting parameter selections are then clearly defined, taking care to describe all details of the formulations.
This will be extremely efficient when numerically solving the initial-boundary-value
problem. This chapter closes by introducing some special matrices, concatenations,
and Kronecker forms which have been used in the Sinc-Galerkin discretization [12].
In Chapter 3, the depth-dependent eddy viscosity is discussed. The model for
a wind-driven current is developed as a complex velocity system and solved by the
Sinc-Galerkin method. The numerical results for this formulation are found in [25].
6
On the other hand, the model is redeveloped in terms of a coupled discrete system
for solving for the real velocity solutions. This Sinc-Galerkin development for the
coupled system has not been done before. The Sinc-Galerkin scheme for the coupled
system is presented and the numerical solutions are reported and then compared to
the numerical solutions of the complex velocity system.
The time-dependent eddy viscosity is introduced in Chapter 4. The wind-driven
current model which involves the time-dependent eddy viscosity is developed in a
partial differential equation. It is transformed to a complex velocity system. The
fully Sinc-Galerkin method in both space and time, which is entirely new for this
oceanography problem, is employed. Several numerical examples for test problems
with known solutions are given to test the performance of the method.
In the final Chapter 5, the above oceanography model is solved, using the ideas
of depth-dependent eddy viscosity in Chapter 3 and time-dependent eddy viscosity
in Chapter 4. These ideas can be combined to determine the numerical solution of
realistic oceanography problems. The problems include both a spin-up wind stress
and an episodic wind stress. These problems are used to show the versatility of the
method and to simulate both velocity components of the ocean current.
7
CHAPTER 2
SINC FUNCTION FUNDAMENTALS
In this chapter, we will review sinc function properties, sinc quadrature rules,
and the Sinc-Galerkin method. These are discussed thoroughly in [12] and [24]. To
develop techniques for solving differential equations, these properties will be used
extensively in Chapter 3 and Chapter 4.
The Sinc Function
The sinc function is defined for all z ∈ C by

sin(πz)


, if z 6= 0
πz
sinc(z) ≡


1
, if z = 0.
For h > 0, the translated sinc function with evenly spaced nodes is given by

sin(π z−kh
)
h

µ
¶ 
, if z 6= kh

z−kh
z − kh
π
h
S(k, h)(z) ≡ sinc
, k = 0, ±1, ±2, ±3, . . . .
≡

h


1
, if z = kh
For various values of k, the sinc basis function S(k, π/4)(x) on the whole real line,
−∞ < x < ∞, is illustrated in Figure 1. For various values of h, the central function
S(0, h)(x) is illustrated in Figure 2.
8
Sinc basis on (−∞, ∞)
1
k = −1
k= 0
k= 1
0.8
S(k, h)(x)
0.6
0.4
0.2
0
-0.2
-0.4
-4
-3
-2
-1
0
1
2
4
3
x
PSfrag replacements
Figure 1. The basis functions S(k, h)(x) for k = −1, 0, 1, with h = π/4.
Sinc basis on (−∞, ∞)
1
h = π/2
h = π/4
h = π/8
0.8
S(0, h)(x)
0.6
0.4
0.2
0
-0.2
-0.4
-4
-3
-2
-1
0
1
2
3
4
x
Figure 2. Central sinc basis function S(0, h)(x) for h = π/2, π/4, π/8.
9
If a function f (x) is defined over the real line, then for h > 0, the series
∞
X
µ
x − kh
C(f, h)(x) =
f (kh)sinc
h
k=−∞
¶
is called the Whittaker cardinal expansion of f whenever this series converges. The
properties of this series have been extensively studied and are thoroughly surveyed in
[12].
The infinite strip, DS of the complex w−plane, where d > 0, is given by
n
πo
DS ≡ w = u + iv :| v |< d ≤
.
2
This infinite strip will be used to discuss and describe those properties.
Sinc Basis Functions for the Galerkin Method
In general, approximations can be constructed for infinite, semi-infinite, and finite
intervals. Both spatial and temporal spaces will be introduced. Define the function
w = φ(z) = ln
µ
z
1−z
¶
,
(2.1)
which is a conformal mapping from DE , the eye-shaped domain in the z−plane, onto
the infinite strip, DS , where
½
DE = z = x + iy
This is shown in Figure 3.
¯
µ
¶¯
¾
¯
¯
z
π
¯
¯
<d≤
.
: ¯arg
1−z ¯
2
10
iy
iv
φ
w−plane
u
x
d
0
1
d
z−plane
DS
DE
φ−1
Figure 3. The relationship between the eye-shaped domain, DE , and the infinite strip,
DS .
For the Sinc-Galerkin method, the basis functions are derived from the composite
translated sinc functions,
µ
¶
φ(z) − kh
,
S(k, h) ◦ φ(z) ≡ sinc
h
for z ∈ DE . These are shown in Figure 4 for real values, x.
(2.2)
The function z =
φ−1 (w) = ew /(1 + ew ) is an inverse mapping of w = φ(z). We may define the range
of φ−1 on the real line as
©
ª
Γ1 = φ−1 (u) ∈ DE : −∞ < u < ∞ .
For the evenly spaced nodes {kh}∞
k=−∞ on the real line, the image which corresponds
to these nodes is denoted by
xk = φ−1 (kh) =
ekh
1 + ekh
(2.3)
11
Sinc basis on (0, 1)
1
k = −1
k= 0
k= 1
0.8
S(k, h) ◦ φ(x)
0.6
0.4
0.2
0
-0.2
-0.4
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 4. Three adjacent members S(k, h) ◦ φ(x) when k = −1, 0, 1 and h =
the mapped sinc basis on the interval (0, 1).
π
8
of
where 0 < xk < 1, for all k. The relationship of the evenly spaced grid in DS to the
sinc grid in DE is shown in Figure 5. The sinc basis functions in (2.2) do not have a
derivative when z tends to 0 or 1. We modify the sinc basis functions as
¡ φ(z) − kh ¢
sinc
S(k, h) ◦ φ(z)
h
≡
φ0 (z)
φ0 (z)
(2.4)
where
1
= z(1 − z).
φ (z)
0
These are shown in Figure 6 for real values, x. Note that the derivatives of the
modified sinc basis functions are defined as z approaches 0 or 1.
12
kh . . .
−4h −3h −2h
xk . . .
PSfrag replacements
−h
0
0
h
2h
.5
3h
...
4h
...
1
R ⊂ DS
(0, 1) ⊂ DE
0.1
Figure 5. The relationship between the grid points xk ∈ (0, 1) and the grid points
kh ∈ R.
Sinc basis on (0, 1)
0.25
k = −1
k= 0
k= 1
0.2
0.2
0
(S(k, h) ◦ φ(x))/φ (x)
0.15
0.1
0.05
0
-0.05
-0.1
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
S(k, h) ◦ φ(x)
when k = −1, 0, 1 and h =
φ0 (x)
the modified sinc basis on the interval (0, 1).
Figure 6. Three adjacent members
π
8
of
13
For the temporal space, we construct an approximation by defining the function
w = Υ(r) = ln(r)
(2.5)
which is a conformal mapping from DW , the wedge-shaped domain in the r−plane
onto the infinite strip, DS , where
n
πo
DW = r = t + is :| arg(r) |< d ≤
.
2
This is shown in Figure 7 For the Sinc-Galerkin method, the basis functions are
is
iv
Υ
w-plane
d
u
t
0
d
DW
r-plane
DS
Υ
−1
Figure 7. The relationship between the wedge-shaped domain, DW , and the infinite
strip, DS .
derived from composite translated functions,
µ
¶
Υ(r) − kh
S(k, h) ◦ Υ(r) ≡ sinc
,
h
for r ∈ DW . These are shown in Figure 8 for real values, t.
(2.6)
The inverse map of
14
Sinc basis on (0, ∞)
1
k = −1
k=0.000
k=1.000
14
0.8
S(k, h) ◦ Υ(t)
0.6
0.4
0.2
0
-0.2
k=
0
k=
1
-0.4
0
2
4
6
8
10
12
t
Figure 8. Three adjacent members S(k, h) ◦ Υ(t) when k = −1, 0, 1 and h =
mapped sinc basis on (0, ∞).
π
8
of the
w = Υ(r) is r = Υ−1 (w) = ew . We may define the range of Υ−1 on the real line as
©
ª
Γ2 = Υ−1 (u) ∈ DW : −∞ < u < ∞ .
For the evenly spaced nodes {kh}∞
k=−∞ on the real line , the image which corresponds
to these nodes is denoted by
tk = Υ−1 (kh) = ekh ,
where 0 < tk < ∞,
for all k.
(2.7)
The relationship of the evenly spaced grid in DS to the sinc grid in DW is shown in
Figure 9. In the remainder of what is done we will choose d = π/2.
15
...
kh
tk
-4h
-3h
...
-2h
0
-h
0
h
1
2h
3h
...
...
R ⊂ DS
(0, ∞) ⊂ DW
Figure 9. The relationship between the grid points tk ∈ (0, ∞) and the grid points
kh ∈ R.
Sinc Interpolation and Quadrature Rules
We develop sinc rules for a special class of functions B(D), where D is an arbitrary
domain. A discussion of the properties of functions in B(D) is found [12] and [24].
Sinc interpolation and quadrature rules for functions in B(D) are also discussed in
[12] and [24]. In particular, this class includes functions such that known exponential
error estimates for these functions exist for infinite sinc interpolation and infinite sinc
quadrature. First we define B(D).
Definition 1. Let φ : D → DS be a conformal map of D to DS with inverse ψ. Let
Γ = {ψ(u) ∈ D : −∞ < u < ∞}. Then B(D) is the class of functions F which are
analytic in D, satisfy
Z
ψ(t+L)
|F (z)|dz → 0 ,
t → ±∞,
16
ª
©
where L = iv :| v |< d ≤ π2 , and on the boundary of D, denoted ∂D, satisfy
Z
N (F ) ≡
∂D
|F (z)|dz < ∞
.
The proofs of the following theorems are found in [12] and [24].
0
Theorem 1. If φ F ∈ B(D) then for all z ∈ Γ,
F (z) =
∞
X
j=−∞
where
µ
πφ(z)
sin
h
EF ≡
2πi
¶
Z
F (zj )S(j, h) ◦ φ(z) + EF
0
∂D
φ (w)F (w)dw
(φ(w) − φ(z)) sin(πφ(w)/h)
.
Theorem 2. If F ∈ B(D) then
Z
∞
X
F (zj )
+ IF
F (z)dz = h
0
φ
(z
)
j
Γ
j=−∞
where
i
IF ≡
2
Z
∂D
F (z)κ(φ, h)(z)
dz
sin(πφ(z)/h)
with
·
¸
iπφ(z)
κ(φ, h)(z) = exp
sgn(I(φ(z))) .
h
The infinite quadrature rule appearing in Theorem 2 can be evaluated directly,
but in general it must be truncated to a finite sum for the Sinc-Galerkin method.
The following theorem indicates the conditions under which exponential convergence
results.
17
Theorem 3. Let F ∈ B(D) and φ be a conformal map with constants α, β, and C so
that

¯
¯
 exp(−α|φ(z)|),
¯ F (z) ¯
¯ 0 ¯≤C
¯ φ (z) ¯

exp(−β|φ(z)|),
where
Γa ≡ {z ∈ Γ : φ(z) = x ∈ (−∞, 0)},
z ∈ Γa
z ∈ Γb
Γb ≡ {z ∈ Γ : φ(z) = x ∈ [0, ∞)}.
Then the sinc trapezoidal quadrature rule is
Z
F (z)dz = h
Γ
N
X
F (zj )
+ O(exp(−αM h)) + O(exp(−βN h))
0
φ
(z
j)
j=−M
+ O(exp(−2πd/h)).
(2.8)
Hence, make the selections
¸
p
¯
¯α
¯
¯
M + 1 , h = 2πd/(αM ),
N=
β
·
where[| · |] denotes the greatest integer, and the exponential order of the sinc trape√
zoidal quadrature rule in (2.8) is O(exp(− 2πdαM )).
Corollary 1. An important special case of Theorem 3 occurs when the integrand has
the form G(z)S(l, h) ◦ φ(z). Due to the interpolation
(0)
S(l, h) ◦ φ(zj ) = S(l, h)(jh) = δjl ,
the sinc quadrature rule is a weighted point evaluation to the order of the method
Z
Γ
G(z)S(l, h) ◦ φ(z)dz = h
G(zl )
+ O(exp(−2πd/h)).
φ0 (zl )
(2.9)
18
In two dimensions, conformal mapping functions w = φ(z) or w = Υ(r) onto intervals of interest will be used and presented in the following corollary. From Theorem
3, the sinc quadrature rule can also provide the approximation of a double integral
as follows.
Corollary 2. For each fixed t, let F (z, t) ∈ B(DE ) and assume there are positive
constants αz , βz , and Cz (t) so that

¯
¯
 exp(−αz |φ(z)|),
¯ F (z, t) ¯
¯ 0
¯ ≤ Cz (t)
¯ φ (z) ¯

exp(−βz |φ(z)|),
where
Γa(z) ≡ {z ∈ Γz : φ(z) = x ∈ (−∞, 0)},
(z)
z ∈ Γa
(z)
z ∈ Γb
(z)
Γb ≡ {z ∈ Γz : φ(z) = x ∈ [0, ∞)}.
Also for each fixed z, let F (z, t) ∈ B(DW ) and assume there are positive constants
αt , βt , and Ct (z) so that

¯
¯
 exp(−αt |Υ(t)|),
¯ F (z, t) ¯
¯ 0
¯ ≤ Ct (z)
¯ Υ (t) ¯

exp(−βt |Υ(t)|),
where
Γ(t)
a ≡ {t ∈ Γt : Υ(t) = x ∈ (−∞, 0)},
(t)
t ∈ Γa
(t)
t ∈ Γb
(t)
Γb ≡ {t ∈ Γt : Υ(t) = x ∈ [0, ∞)}.
Then the sinc trapezoidal quadrature rule is
Z Z
Γt
F (z, t)dzdt = hz ht
Γz
Nz
X
Nt
X
j=−Mz k=−Mt
F (zj , tk )
φ (zj )Υ0 (tk )
0
+ O(exp(−αz Mz hz )) + O(exp(−βz Nz hz ))
+ O(exp(−2πd/hz )) + O(exp(−αt Mt ht ))
+ O(exp(−βt Nt ht )) + O(exp(−2πd/ht )).
(2.10)
19
Hence, make the selections
Nz =
·
¸
·
¸
·
¸
¯ αz
¯
¯
¯
¯
¯
¯ Mz + 1 ¯ , M t = ¯ α z Mz + 1 ¯ , N t = ¯ α z Mz + 1 ¯ ,
βz
αt
βt
where h = hz = ht , and
p
h =
2πd/(αz Mz ),
√
and the exponential order of the sinc trapezoidal quadrature rule is O(exp(− 2πdαz Mz )).
Corollary 3. An important special case housed in Corollary 2 occurs when the double
integrand has the form G(z, t)S(p, hz ) ◦ φ(z)S(q, ht ) ◦ Υ(t). Due to the interpolation
(0)
S(p, hz ) ◦ φ(zj ) = S(p, hz )(jhz ) = δjp
(0)
and S(q, ht ) ◦ Υ(tk ) = S(q, ht )(kht ) = δkq ,
the sinc quadrature rule is a weighted point evaluation to the order of the method
Z Z
Γt
Γz
G(z, t)S(p, hz ) ◦ φ(z)S(q, ht ) ◦ Υ(t)dzdt = hz ht
G(zj , tk )
φ (zj )Υ0 (tk )
0
+O(exp(−2πd/hz )) + O(exp(−2πd/ht )) .
(2.11)
The General Sinc-Galerkin Method
Consider an ordinary differential equation
Ly = f
(2.12)
on Γ. In the Sinc-Galerkin method we assume an approximate solution ym in the
form of a series with m = M + N + 3 terms,
ym (z) =
N
+1
X
j=−M −1
yj S(j, h) ◦ φ(z).
20
© ªN +1
The coefficients yj j=−M −1 are determined by orthogonalizing the residual Lym − f
© ªN +1
with respect to the sinc basis functions Sk k=−M −1 where Sk (z) ≡ S(k, h) ◦ φ(z).
The inner product
(F, G) =
Z
F (z)G(z)w(z)dz,
Γ
which uses a weight function w chosen depending on the boundary conditions, is
implemented in the orthogonalization. So this yields the discrete Galerkin system
Z
Γ
(Lym − f )(z)S(k, h) ◦ φ(z)w(z)dz = 0 ,
for − M − 1 ≤ k ≤ N + 1. (2.13)
If the form in (2.12) becomes a partial differential equation, then we also assume
an approximate solution ymz ,mt in the form of a series with mz = Mz + Nz + 3 , mt =
Mt + Nt + 2 terms,
ymz ,mt (z, t) =
N
z +1
X
N
t +1
X
j=−Mz −1 k=−Mt
The coefficients
©
cjk
ªNz +1,Nt +1
j=−Mz −1,k=−Mt
cjk S(j, h) ◦ φ(z)S(k, h) ◦ Υ(t).
which can form an mz × mt matrix C = [cjk ]
are determined by orthogonalizing the residual Lymz ,mt − f with respect to the sinc
©
ªNz +1,Nt +1
basis functions Sp Sq∗ p=−Mz −1,q=−Mt where Sp Sq∗ (z, t) ≡ S(p, h) ◦ φ(z)S(q, h) ◦ Υ(t)
for −Mz − 1 ≤ p ≤ Nz + 1, −Mt ≤ q ≤ Nt + 1. The inner product used is
(F, G) =
Z Z
Γt
F (z, t)G(z, t)$(z, t)dzdt.
Γz
The choice of the weight function $(z, t) in the double integrand depends on the
boundary conditions, the domain, and the partial differential equation. Therefore the
21
discrete Galerkin system is
Z Z
Γt
Γz
(Lymz ,mt − f )(z, t)Sp Sq∗ (z, t)$(z, t)dzdt = 0.
(2.14)
Matrix Representation of the Derivatives of Sinc Basis Functions at Nodal Points
The Sinc-Galerkin method actually requires the evaluated derivatives of sinc basis
functions S(p, h) ◦ φ(z) at the sinc nodes, z = zj . The rth derivative of S(p, h) ◦ φ(z),
with respect to φ, evaluated at the nodal point zj is denoted by
¯
dr
1 (r)
≡
δ
[S(p, h) ◦ φ(z)] ¯z=zj .
pj
r
r
h
dφ
(2.15)
(r)
The expressions in (2.15) for each p and j can be stored in a matrix I (r) = [δpj ].
For r = 0, 1, 2,
(0)
(0)
I (0) = [δpj ] , where δpj
I
I
(1)
(2)
=
=
(1)
[δpj ]
(2)
[δpj ]
, where
, where
(1)
δpj
(2)
δpj
≡ [S(p, h) ◦ φ(z)]|z=zj =

 1 , if j = p

0 , if j 6= p ,

 0
d
≡ h [S(p, h) ◦ φ(z)]|z=zj =

dφ
, if j = p
(−1)j−p
j−p



d2
≡ h2 2 [S(p, h) ◦ φ(z)]|z=zj =

dφ

, if j 6= p ,
−π 2
3
, if j = p
−2(−1)j−p
(j−p)2
, if j 6= p.
The following matrices will be some examples for I (0) , I (1) , I (2) . Given −Mz − 1 ≤
p ≤ Nz + 1 and −Mz − 1 ≤ j ≤ Nz + 1, (mz = Mz + Nz + 3), the mz × mz , square
22
matrices I (0) , I (1) , I (2) are given by

1 ...
 .. . .
= .
.
0 ...
I (0)

I (1)
0

 1


=  −1
2

 ..
 .
(−1)mz
mz −1

I (2)




= 



2
− π3
2
− 222
..
.
−2(−1)mz −1
(mz −1)2

0
..  = I,
.
(2.16)
1
(−1)mz −1
mz −1
−1 21 . . .
... ... ...
... ... ...
... ... ...
. . . − 12 1
..
.
1
2
−1
0
2 − 222 . . .
... ... ...
... ... ...
... ... ...
. . . − 222 2





,



−2(−1)mz −1
(mz −1)2
..
.
− 222
2
2
− π3
(2.17)





.



(2.18)
When −Mz ≤ j ≤ Nz , we remove the first and last columns of I (0) , I (1) , and I (2) in
(2.16)-(2.18), to arrive at the mz × nz , (nz = Mz + Nz + 1), non-square matrices


0 ... 0
1 . . . 0 




(2.19)
Iz(0) =  ... . . . ...  ,


0 . . . 1 
0 ... 0

Iz(1)





= 





−1
0
1
− 12
..
.
(−1)mz −1
mz −2
1
2
...
... ...
... ...
... ...
...
...
...
− 12
(−1)mz −2
mz −2
..
.
1
2
−1
0
1






,





(2.20)
23






= 





Iz(2)
− 222
...
...
...
...
...
...
..
.
...
...
−2(−1)mz −2
...
− 222
2
−π 2
3
2
22
− 322
(mz −2)2
−2(−1)mz −2
(mz −2)2
..
.
...
− 322
2
22
−π 2
3
2






.





(2.21)
When −Mt ≤ q ≤ Nt +1 and −Mt ≤ j ≤ Nt , (mt = M t+Nt +2 and nt = M t+Nt +1),
we remove the last column of I (0) , I (1) , and I (2) in (2.16)-(2.18), to leave the mt × nt
non-square matrices

1
 ..

= .
0
0
(0)
It

(1)
It
0

 1


=  −1
 2
 ..
 .
(−1)mt
mt −1

(2)
It




= 



2
− π3
2
− 222
..
.
−2(−1)mt −1
(mt −1)2

... 0
. . . .. 
.
,
. . . 1
... 0
−1 . . .
... ...
... ...
... ...
...
− 12
2
...
...
...
...
...
...
...
. . . − 222
(2.22)
(−1)mt −2
mt −2
..
.
−1
0
1





,



−2(−1)mt −2
(mt −2)2
..
.
2
22
2
− π3
2
(2.23)





.



(2.24)
If a function f is evaluated at the sinc nodes z = zj for −Mz ≤ j ≤ Nz , then the
nz × nz square diagonal matrix Dnz (f ) is written by


0
f (z−Mz )
...






f (z0 )
Dnz (f ) = 
.


.
..


0
f (zNz )
(2.25)
24
The chain rule has been used for the z-derivative of product sinc functions. For
example, when Sj (z) = S(j, h) ◦ φ(z),
¶
dω(z)
dSj (z) dφ(z)
ω(z) + Sj (z)
dφ(z) dz
dz
dSj (z) 0
0
=
φ (z)ω(z) + Sj (z)ω (z)
dφ
d(Sj (z)ω(z))
=
dz
µ
(2.26)
and
µ
¶
d2 (Sj (z)ω(z))
d dSj (z) 0
0
=
φ (z)ω(z) + Sj (z)ω (z)
dz 2
dz
dφ
dSj (z) 00
dSj (z) 0
d2 Sj (z) 0
0
(φ (z))2 ω(z) +
φ (z)ω(z) + 2
φ (z)ω (z)
=
2
dφ
dφ
dφ
00
+ Sj (z)ω (z)
µ
¶
dSj (z) 00
d2 Sj (z) 0
0
0
2
φ (z)ω(z) + 2φ (z)ω (z)
(φ (z)) ω(z) +
=
dφ2
dφ
00
+ Sj (z)ω (z).
In the special case where φ(z) = ln
(2.27)
µ
¶
z
1
, ω(z) = √ 10 and 0
= z(1 − z),
φ (z)
1−z
φ (z)
then
1
p
φ0 (z)
µ
1
p 0
φ (z)
¶0
1
= (1 − 2z) and
2
From (2.26), (2.27) we get
µ
¶00
1
1
1
p 0
=− .
0
3/2
(φ (z))
4
φ (z)
µ
¶0
d(Sj (z)ω(z))
1
dSj (z) p 0
=
φ (z) + Sj (z) p 0
dz
dφ
φ (z)
µ
¶
p 0
dSj (z) 1
=
+ (1 − 2z)Sj (z)
φ (z)
dφ
2
(2.28)
25
and
µ 00
¶
00
d2 (Sj (z)ω(z))
d2 Sj (z) 0
dSj (z) φ (z)
−φ (z)
0
3/2
p 0
=
(φ (z)) +
+ 2φ (z) ¡ 0 ¢3/2
dz 2
dφ2
dφ
φ (z)
2 φ (z)
µ
¶00
1
+ Sj (z) p 0
φ (z)
¶
µ 2
d Sj (z) 1
0
− Sj (z) (φ (z))3/2 .
(2.29)
=
2
dφ
4
p
1
= z, then
If φ(z) = ln(z), ω(z) = φ0 (z), and 0
φ (z)
1
00
φ (z) = −1.
2
(φ (z))
0
From (2.26), we get
0
d(Sj (z)ω(z))
dSj (z) 0
φ (z)
=
(φ (z))3/2 + Sj (z) p 0
dz
dφ
2 φ (z)
¶
µ
dSj (z) 1
0
− Sj (z) (φ (z))3/2 .
=
dφ
2
(2.30)
Matrices and Kronecker Products
We will develop the notation and tools to formulate discrete systems for complex
problems. These tools are described in [6] and [12].
Definition 2. For an m × n matrix B = [bij ],
£
¤
B = b1 b2 . . . b n ,
1 ≤ i ≤ m, 1 ≤ j ≤ n,
 
b1j
 b2j 
 
where bj =  ..  ,
 . 
bmj
the concatenation of B is the mn × 1 vector
 
b1
 b2 
 
co(B) =  ..  .
.
bn
(2.31)
26
Definition 3. Let A be an m × n matrix and B be a p × q matrix. The Kronecker or
tensor product of A and B is an mp × nq matrix


. . . a1n B
. . . a2n B 

..  .
. 
am1 B am2 B . . . amn B
a11 B
 a21 B

A ⊗ B =  ..
 .
a12 B
a22 B
..
.
(2.32)
Useful properties of concatenation are given in the following theorems.
Theorem 4. If A and B are matrices of identical dimension and α and β are scalars
then
co(αA + βB) = αco(A) + βco(B).
Theorem 5. Let A be m × m, X be m × n, and B be n × n. Then
co(AXB) = (B T ⊗ A)co(X).
Theorem 6. Let the linear system for the unknown matrix X be given as
A1 XB 1 + A2 XB 2 + . . . + Ak XB k = C
(2.33)
where Ai are m × m, X, C are m × n, and B i are n × n. This is equivalent to
Gco(X) = co(C)
where G ≡ B T1 ⊗ A1 + B T2 ⊗ A2 + . . . + B Tk ⊗ Ak .
(2.34)
27
CHAPTER 3
WIND-DRIVEN CURRENTS IN A SEA WITH A DEPTH-DEPENDENT EDDY
VISCOSITY
In this chapter, we will perform a numerical study on a model of wind-driven
currents in open and semi-enclosed seas. This model is found in the work of Winter,
Bowers, and Lund [25]. In their research, they applied the Sinc-Galerkin technique
to solve a model describing wind-driven subsurface currents in coastal regions and
semi-enclosed seas when the vertical eddy viscosity coefficient is represented as a
continuously differentiable function of depth.
Governing Equations
To develop a mathematical model, we first construct a right-handed coordinate
system with the vertical coordinate z ∗ directed positive downward from the free surface, and with x∗ and y ∗ directed northward and eastward, respectively. A plane
at z ∗ = D0 corresponds to the impermeable boundary at the seabed. For the purpose of illustrating the Sinc-Galerkin method, we employ several of the simplifying
assumptions invoked in one-dimensional wind-drift studies - the ocean depth, D0 ,
and mass density, ρ, are assumed constant, and the effects of tides, inertial terms,
free surface slope, and variations in atmospheric pressure are neglected. Currents are
driven by a tangential surface wind stress (the surface is at z ∗ = 0) of magnitude
28
x∗
q ∗ (z ∗ )
Sea surface
χ
y∗
z =0
∗
τ (0) = τw (cos(χ)x̂∗ + sin(χ)ŷ∗ )
τ (z ∗ ) = −ρA∗v (z ∗)dq ∗ /dz ∗
A∗v (z ∗ ), eddy viscosity
z∗
z ∗ = D0 , Seabed
Figure 10. The general physical model of the depth-dependent eddy viscosity oceanography problem.
τw represented as τ (0) = τw (cos(χ)x̂∗ + sin(χ)ŷ ∗ ), with χ being the angle between
the positive x∗ -axis and the wind direction in Figure 10.
Here x̂∗ and ŷ ∗ repre-
sent unit vectors in the directions of the positive x∗ -axis and the positive y ∗ -axis,
respectively. In this model, called a specified eddy viscosity model, internal frictional
stresses are parameterized as τ (z ∗ ) = −ρA∗v (z ∗ )dq ∗ /dz ∗ , where the specified effective
vertical eddy viscosity coefficient A∗v (z ∗ ) is a continuously differentiable function of
z ∗ ∈ (0, D0 ) and ρ is the ocean mass density. Here q ∗ (z ∗ ) = U ∗ (z ∗ )x̂∗ + V ∗ (z ∗ )ŷ ∗
represents the horizontal wind-drift current which is the difference between the total
velocity and the geostrophic current in [3]. The Coriollis force acting on the moving
water is balanced by a horizontal pressure gradient force.
29
Under the present assumptions, the conservation of linear momentum equations
express a balance between the Coriollis force and the internal friction associated with
turbulence. The wind-drift current q ∗ is determined by solving the boundary-value
problem
d
dz ∗
µ
dq ∗
A∗v (z ∗ ) ∗
dz
¶
= −f ẑ ∗ × q ∗ , 0 < z ∗ < D0 ,
(3.1)
where the stress condition at the sea surface, z ∗ = 0, is the tangential surface wind
stress
−ρA∗v (0)
dq ∗ (0)
= τw (cos(χ)x̂∗ + sin(χ)ŷ ∗ )
dz ∗
(3.2)
while at the seabed, z ∗ = D0 , the frictional stress is linearly proportional to the
current, hence
−ρA∗v (D0 )
dq ∗ (D0 )
= kf ρq ∗ (D0 ).
dz ∗
(3.3)
Here f ≡ 2Ω sin θ is the Coriollis parameter at latitude θ where Ω = 7.29×10−5 rad s−1
is the angular speed of rotation of the earth. Though −π/2 < θ < π/2 , for the rest
of our work we assume we are in the northern hemisphere and hence θ > 0 (f > 0).
Here kf is the linear slip bottom stress coefficient. If q ∗ (D0 ) = 0, (3.3) is called a
zero-velocity (no-slip) condition.
From (3.1), we can simplify the boundary-value problem as follows
d
dz ∗
µ
dq ∗
A∗v (z ∗ ) ∗
dz
¶
d
=
dz ∗
µ
(z ∗ )
dz ∗
dU
A∗v (z ∗ )
∗
¶
d
x̂ + ∗
dz
∗
µ
(z ∗ )
dz ∗
dV
A∗v (z ∗ )
∗
¶
ŷ ∗
30
= −f ẑ ∗ × q ∗
= −f ẑ ∗ × [U ∗ (z ∗ )x̂∗ + V ∗ (z ∗ )ŷ ∗ ]
¡
¢
= −f U ∗ (z ∗ )ŷ ∗ − V ∗ (z ∗ )x̂∗ , 0 < z ∗ < D0 .
Then it can be written in component form as
d
− ∗
dz
µ
d
− ∗
dz
µ
(z ∗ )
dz ∗
dU
A∗v (z ∗ )
∗
¶
= −f V ∗ (z ∗ ), 0 < z ∗ < D0
(3.4)
= f U ∗ (z ∗ ), 0 < z ∗ < D0 .
(3.5)
and
(z ∗ )
dz ∗
dV
A∗v (z ∗ )
∗
¶
The stress condition at the sea surface, z ∗ = 0, separates as
−ρA∗v (0)
dU ∗ (0)
= τw cos(χ) ,
dz ∗
−ρA∗v (0)
dV ∗ (0)
= τw sin(χ).
dz ∗
(3.6)
At the seabed, z ∗ = D0 , the frictional stress separates as
−ρA∗v (D0 )
dU ∗ (D0 )
= kf ρU ∗ (D0 ) ,
dz ∗
−ρA∗v (D0 )
dV ∗ (D0 )
= kf ρV ∗ (D0 ).
dz ∗
(3.7)
To nondimensionalize the model equations we begin with a measure of near surface
turbulent eddy viscosity, A0 ≡ A∗v (0), and define a nominal “upper-layer” Ekman
depth by DE ≡
√
p
2A0 /f . Also define a current speed in units of U0 = τw DE /(ρA0 ) =
√
2τw /(ρ A0 f ). U0 is the natural velocity scale in an infinitely deep sea with uniform
eddy viscosity in the steady-state. A nondimensional form of the equations of motion
can be expressed with the introduction of the nondimensional variables
z≡
z∗
A∗ (z ∗ )
q ∗ (z ∗ )
, Av (z) ≡ v∗
, q(z) ≡
≡ U (z)x̂ + V (z)ŷ
D0
Av (0)
U0
(3.8)
31
together with two nondimensional constants (a depth ratio κ and a bottom friction
parameter σ) given by
D0
= D0
κ≡
DE
r
f
,
2A0
σ≡
A0 Av (1)
A∗ (D0 )
= v
.
k f D0
k f D0
(3.9)
We apply (3.8) and (3.9) to nondimensionalize the equations (3.4) and (3.5). For
(3.4), we have
1 d
−
D0 dz
µ
1 d(U0 U (z))
A0 Av (z)
D0
dz
¶
= −f U0 V (z), 0 < z < 1,
which results in
d
−
dz
µ
dU (z)
Av (z)
dz
¶
= −2κ2 V (z), 0 < z < 1.
(3.10)
Similarly, the result from (3.5) is
d
−
dz
µ
dV (z)
Av (z)
dz
¶
= 2κ2 U (z), 0 < z < 1.
(3.11)
The surface boundary conditions in (3.6) are nondimensionalized to produce
−ρ
A0 d(U0 U (0))
= τw cos(χ)
D0
dz
A0 d(U0 V (0))
= τw sin(χ),
D0
dz
,
−ρ
,
dV (0)
= −κ sin(χ).
dz
which leads to
dU (0)
= −κ cos(χ)
dz
(3.12)
The seabed conditions in (3.7) are also nondimensionalized and become
−
A0 Av (1) d(U0 U (1))
= kf U0 U (1)
D0
dz
,
−
A0 Av (1) d(U0 V (1))
= kf U0 V (1),
D0
dz
32
which leads to
U (1) + σ
dU (1)
=0
dz
,
V (1) + σ
dV (1)
= 0.
dz
(3.13)
In the special no-slip case, we can set σ = 0 in (3.13).
We first transform the nonhomogenous boundary conditions to homogeneous
boundary conditions by using the linear transformations
U (z) = u(z) + κ (1 + σ − z) cos(χ),
V (z) = v(z) + κ (1 + σ − z) sin(χ).
(3.14)
The first derivative of each transformation in (3.14) yields
du(z)
dU (z)
=
− κ cos(χ),
dz
dz
dV (z)
dv(z)
=
− κ sin(χ),
dz
dz
so the resulting boundary-value problem for u(z) satisfies
d
−
dz
µ
¶
du
d
Av (z)
+
(Av (z)κ cos(χ)) = −2κ2 (v(z) + κ (1 + σ − z) sin(χ)),
dz
dz
0 < z < 1,
which gives
d
−
dz
µ
¶
du
Av (z)
+ κ cos(χ)A0v (z) = −2κ2 v(z) − 2κ3 (1 + σ − z) sin(χ),
dz
0 < z < 1.
(3.16)
Similarly the boundary-value problem for v(z) satisfies
d
−
dz
µ
dv
Av (z)
dz
¶
+
d
(Av (z)κ sin(χ)) = 2κ2 (u(z) + κ (1 + σ − z) cos(χ)),
dz
0 < z < 1,
33
which gives
d
−
dz
µ
dv
Av (z)
dz
¶
+ κ sin(χ)A0v (z) = 2κ2 u(z) + 2κ3 (1 + σ − z) cos(χ),
0 < z < 1.
(3.17)
dv(0)
=0
dz
(3.18)
The boundary conditions (3.12) become
du(0)
= 0,
dz
while the seabed boundary conditions (3.13) become
u(1) + σ
du(1)
= 0,
dz
v(1) + σ
dv(1)
= 0.
dz
(3.19)
For the purpose of illustrating the exposition of the Sinc-Galerkin technique, we define
d
Lu(z) ≡ −
dz
µ
du
Av (z)
dz
¶
,
d
Lv(z) ≡ −
dz
µ
dv
Av (z)
dz
¶
.
(3.20)
Then (3.16) and (3.17) are now given by the coupled u and v equation systems
Lu(z) + 2κ2 v(z) = F1 (z),
0 < z < 1,
(3.21)
Lv(z) − 2κ2 u(z) = F2 (z),
0 < z < 1,
(3.22)
and
where
0
F1 (z) = −2κ3 (1 + σ − z) sin(χ) − κ cos(χ)Av (z)
and
0
F2 (z) = 2κ3 (1 + σ − z) cos(χ) − κ sin(χ)Av (z).
34
The surface boundary conditions are
du(0)
= 0,
dz
dv(0)
= 0.
dz
(3.23)
The seabed boundary conditions are
u(1) + σ
du(1)
= 0,
dz
v(1) + σ
dv(1)
= 0.
dz
(3.24)
Another way to address (3.21)-(3.24) is to introduce a complex velocity and
formulate the Sinc-Galerkin technique in the complex plane. First multiply (3.22)
by the imaginary unit i, add the result to (3.21) , and define a complex velocity
w(z) = u(z) + iv(z). If one defines
Lw(z) ≡ Lu(z) + iLv(z)
µ
¶
µ
¶
d
du(z)
d
dv(z)
≡ −
Av (z)
−i
Av (z)
dz
dz
dz
dz
µ
¶
dw(z)
d
≡ −
Av (z)
,
dz
dz
(3.25)
the problems in (3.21) and (3.22) are now given by
Lw(z) − i2κ2 w(z) = F (z), 0 < z < 1,
(3.26)
where
F (z) ≡ 2κ3 [i(1 + σ − z) cos(χ) − (1 + σ − z) sin(χ)]
0
− κAv (z) [cos(χ) + i sin(χ)]
=
£
¤
−κA0v (z) + i2κ3 (1 + σ − z) eiχ .
(3.27)
35
Repeating this procedure with (3.23), the surface condition becomes
w0 (0) = 0,
(3.28)
and with (3.24), the seabed condition becomes
w(1) + σw 0 (1) = 0.
(3.29)
Sinc-Galerkin Solution of the Complex Velocity Formulation
The Sinc-Galerkin procedure for the complex velocity formulation in (3.26)-(3.29)
begins by selecting composite sinc functions appropriate to the interval (0, 1) as the
basis functions for the expansion of approximate solutions for the current components,
w(z). With the introduction of the conformal mapping function in (2.1), φ(z) =
ln(z/(1 − z)), the appropriate composite sinc functions in (2.2), S(j, h) ◦ φ(z), over
the interval z ∈ (0, 1) are given by
¡π
¢

sin
(φ(z)
−
jh)

h
¶ 
µ
, φ(z) 6= jh

π
φ(z) − jh
(φ(z)
−
jh)
h
≡
S(j, h) ◦ φ(z) ≡ sinc

h


1
, φ(z) = jh
(3.30)
for j = −N, −N + 1, . . . , N − 1, N .
When the boundary conditions are Dirichlet conditions, the representations of the
solution are provided in the immediate neighborhood of the endpoints by using the
composite functions in (3.30). These sinc functions tend to zero and their derivatives
become undefined as z approaches 0 or 1. For Neumann or radiation boundary
conditions a modification is necessary for the composite sinc functions. So we use
36
(2.4) and add boundary basis functions that are Hermite polynomials. Thus the
derivatives at z = 0 and 1 for these basis functions are defined. These Hermite
polynomials are given by
B0 (z) = (2z + 1)(1 − z)2 , B1 (z) = (1 − γz)z 2 where γ ≡
1 + 2σ
.
1 + 3σ
(3.31)
The approximate solution for w(z) in (3.26), subject to the mixed conditions
(3.28) and (3.29), is represented by the expansion
wa (z) = c−N −1 B0 (z) +
N
X
cj
j=−N
S(j, h) ◦ φ(z)
+ cN +1 B1 (z),
φ0 (z)
(3.32)
where the expansion weight function, 1/φ0 (z) = z(1 − z) has been introduced as in
(2.4). It remains to determine the m = 2N + 3 coefficients in the expansion wa (z) of
(3.32) that approximates w(z).
The strategy for carrying out this task is outlined below. The result is a straightforward linear matrix equation for the unknown coefficients in the sinc function expansion (3.32).
First substitute wa (z) in the governing differential equation (3.26) to arrive at
Lwa (z) − i2κ2 wa (z) − F (z) = 0,
0 < z < 1,
(3.33)
where we write
wa (z) = wh (z) + wb (z).
(3.34)
Here
wh (z) =
N
X
j=−N
cj
S(j, h) ◦ φ(z)
φ0 (z)
(3.35)
37
refers to the contribution to wa (z) associated with the homogeneous Dirichlet problem
in the sinc expansion in (3.32). Also
wb (z) = c−N −1 B0 (z) + cN +1 B1 (z)
(3.36)
is the contribution to wa (z) associated with the boundary conditions. From (3.35)
0
0
and (3.36) we compute wh (z) and wb (z) and evaluate them at the endpoints as
0
0
wh (0) = 0 = wh (1) ,
0
0
wb (1) = −cN +1
wb (0) = 0 ,
1
,
1 + 3σ
and
wh (0) = 0 = wh (1) ,
wb (0) = c−N −1
,
wb (1) = cN +1
σ
.
1 + 3σ
By using these results, we can show that wa (z) satisfies the boundary conditions at
the surface
0
0
0
wa (0) = wb (0) + wh (0) = 0,
(3.37)
and at the seabed
0
0
0
wa (1) + σwa (1) = wb (1) + wh (1) + σwb (1) + σwh (1) = 0.
(3.38)
Orthogonalize the residual with respect to the set of composite sinc functions
S(p, h) ◦ φ(z), with inner product weight ω(z), for p = −N − 1, −N, . . . , N, N + 1.
Thus we have
Z
0
1£
¤
Lwa (z) − i2κ2 wa (z) − F (z) S(p, h) ◦ φ(z)ω(z)dz = 0
38
for p = −N − 1, −N, . . . , N, N + 1, which becomes
Z
1
0
Lwh (z)S(p, h) ◦ φ(z)ω(z)dz
(3.39)
Z 1
£
¤
+
Lwb (z) − i2κ2 wa (z) − F (z) S(p, h) ◦ φ(z)ω(z)dz = 0.
0
The integral involving Lwh (z) requires special treatment and the remaining integral
can be accurately estimated using the special case of the sinc trapezoidal quadrature
rule in (2.9).
Now denote the first integral in (3.39) by J, and approximate it by first integrating
by parts twice to transfer derivatives from wh (z) to S(p, h) ◦ φ(z)ω(z). During this
process, some clarity is achieved if S(p, h) ◦ φ(z) is abbreviated by Sp (z). Following
two integrations by parts, J becomes
J =
·
¶
¸1
dSp 0
φ ω (z)Av (z)wh (z)
+ Sp ω +
dφ
0
Z 1
£
¤0
−
Av (z) (Sp (z)ω(z))0 wh (z)dz.
− (Sp ωAv wh0 ) (z)
µ
0
(3.40)
0
With regard to the first endpoint contribution, from physical considerations it is
reasonable to assume that the reduced velocity shear w 0 (z) remains bounded as z
approaches 0 and 1. Thus as z −→ 0, wh (z) is O (z α ) , α ≥ 1, and similarly, as
¡
¢
z −→ 1, wh (z) is O (1 − z)β , β ≥ 1. The inner product weight assignment ω(z) =
(φ0 (z))−1/2 =
p
z(1 − z) is adequate to nullify the first endpoint contribution. In
the second endpoint term, the chain rule has been used to rewrite the z-derivative of
Sp (z) as φ0
dS(p, h) ◦ φ(z)
. As z approaches 0 and 1 the first derivative of the sinc
dφ
function is not defined. However, since wh (z) is expanded in terms of composite sinc
39
functions with expansion weight 1/φ0 (z) = z(1 − z), the same choice of inner product
weight, ω(z) = (φ0 (z))−1/2 =
p
z(1 − z) leads to the inequalities
|Sp (z)ω 0 (z)Av (z)wh (z)| ≤ C1
p
z(1 − z)
and
¯
¯
p
¯
¯ dSp
0
¯
¯
≤
C
(z)φ
(z)ω(z)A
(z)w
(z)
z(1 − z).
2
v
h
¯
¯ dφ
(3.41)
Hence we can be assured that both boundary contributions are zero by choosing
ω(z) = (φ0 (z))−1/2 =
p
z(1 − z) as the inner product weight function. After carrying
out the operations in the integrand in (3.40), including evaluations of the derivative
combinations of the mapping φ(z) = ln(z/(1 − z)), the integral J can be expressed
as
J =−
Z
1
0
h
i
0
(Sp (z)ω(z))00 Av (z) + (Sp (z)ω(z))0 Av (z) wh (z)dz.
Apply the z-derivative of the sinc products in (2.26) and (2.27), and use the special
cases in (2.28) and (2.29) to arrive at
1
½ 2
¾
d Sp (z) 1
3/2
J =
−
+ Sp (z) Av (z) (φ0 (z)) wh (z)dz
2
dφ
4
¾
Z0 1 ½
dSp (z) 1
1/2
−
+
+ (2z − 1) Sp (z) A0v (z) (φ0 (z)) wh (z)dz.
dφ
2
0
Z
(3.42)
With J represented by (3.42), after some rearrangement of terms, (3.39) can be
written in a form convenient for quadrature estimation
40
c−N −1
Z
1
0
0
{LB0 (z)} Sp (z) (φ (z))
−1/2
dz + J − i2κ
2
Z
1
Sp (z) (φ0 (z))
−1/2
wa (z)dz
0
(3.43)
+cN +1
Z
1
0
0
{LB1 (z)} Sp (z) (φ (z))
−1/2
dz =
Z
1
F (z)Sp (z) (φ0 (z))
−1/2
dz.
0
The integrals J in (3.42) are approximated using the sinc trapezoidal quadrature rule
in (2.8),
Z
1
0
N
X
G(zj )
.
G(z)dz ≈ h
φ0 (zj )
j=−N
(3.44)
Here zj = ejh /(ejh + 1) are the nodal points corresponding to a uniform grid of jh
operated on by the inverse of the mapping function φ. The integrals in (3.43) use the
special case of the sinc quadrature rule in Corollary 1,
Z
1
0
G(z)S(p, h) ◦ φ(z)dz ≈ h
G(zp )
.
φ0 (zp )
(3.45)
Consider first the approximation of the integral J and begin by applying (3.44)
to the integrals involving the φ-derivatives of Sp . These are readily evaluated using
(r)
δpj , which represents the r th derivative of S(p, h) ◦ φ(z) with respect to φ evaluated
at zj . Use of these representations leads to the final approximation for J (using the
nodal points zj )
N ½
X
−1 (2)
δ +
J ≈ h
h2 pj
j=−N
N ½
X
−1 (1)
+ h
δpj +
h
j=−N
¾
1 (0)
1/2
(3.46)
δpj Av (zj ) (φ0 (zj )) wh (zj )
4
¾
1
−1/2
(0)
(2zj − 1) δpj A0v (zj ) (φ0 (zj ))
wh (zj ) ,
2
41
for p = −N − 1, −N, . . . , N, N + 1. From (3.35) and the interpolation property of
the sinc basis, one has wh (zj ) = cj /φ0 (zj ) which, in conjunction with (3.45), provides
the approximation of the remaining integrals in (3.43). Using (3.46) in this result,
one arrives at the following system of m = 2N + 3 linear equations whose solution
gives the coefficients of the expansion (3.32) for wa (z).
For p = −N − 1, −N, . . . , 0, . . . , N, N + 1,
¾
N ½
X
−1 (2) 1 (0)
LB0 (zp )
−1/2
c−N −1 +
δpj + δpj Av (zj ) (φ0 (zj ))
cj
0
3/2
2
(φ (zp ))
h
4
j=−N
¾
N ½
X
−1 (1) 1
−3/2
(0)
+
δpj + (2zj − 1) δpj A0v (zj ) (φ0 (zj ))
cj
h
2
j=−N
+
(3.47)
F (zp )
LB1 (zp )
cN +1 − i2κ2 (φ0 (zp ))−3/2 wa (zp ) = 0
.
0
3/2
(φ (zp ))
(φ (zp ))3/2
The numerators in the boundary terms of (3.47) are
d
LB0 (z) = −
dz
µ
¶
µ
¶
dB0 (z)
d
Av (z)
= 6
Av (z)z(1 − z)
dz
dz
¶
µ
0
= 6 Av (z)(1 − 2z) + Av (z)z(1 − z)
and
µ
¶
µ
¶
d
dB1 (z)
d
LB1 (z) = −
Av (z)
= −
Av (z)(2 − 3γz)z
dz
dz
dz
µ
¶
0
= − 2Av (z)(1 − 3γz) + Av (z)(2 − 3γz)z .
Since (3.47) is a linear system for the expansion coefficients, it is advantageous to
rewrite it using matrix notation. Introduce the column vectors
c ≡ [c−N −1 c−N · · · cN cN +1 ]T ,
42
wa ≡ [wa (z−N −1 ) wa (z−N ) · · · wa (zN ) wa (zN +1 )]T ,
and
F ≡ [F (z−N −1 ) F (z−N ) · · · F (zN ) F (zN +1 )]T ,
where wa (z) is given in (3.32) and F (z) is given in (3.27). Remember m = 2N +3 and
define n = 2N + 1. For any function g, the n × n square diagonal matrix Dn (g) has
(0)
(1)
diagonal entries g(z−N ), · · · , g(zN ). The elements of the m × n matrices Iz , Iz
(2)
and Iz
(0)
(1)
(2)
in (2.19)-(2.21) are the sinc delta functions δpj , δpj , and δpj , respectively.
The form of (3.47), for p = −N − 1, −N, . . . , N, N + 1, suggests the definition of
the non-square m × n matrix
Ans
Ã
!
¾
Av
−1 (2) 1 (0)
Dn
I + Iz
≡
h2 z
4
(φ0 )1/2
Ã
!
½
¾
0
−1 (1) 1 (0)
Av
+
I + Iz Dn (2z − 1) Dn
,
h z
2
(φ0 )3/2
½
(3.48)
the m × 1 column vectors a−N −1 and aN +1 with pth component,
p = −N − 1, −N, . . . , 0, . . . , N, N + 1,
[a−N −1 ]p ≡
LB0 (zp )
,
(φ0 (zp ))3/2
[aN +1 ]p ≡
LB1 (zp )
,
(φ0 (zp ))3/2
(3.49)
and the m × 1 column vectors b−N −1 and bN +1 with pth component
[b−N −1 ]p ≡ B0 (zp ) ,
[bN +1 ]p ≡ B1 (zp ).
(3.50)
To build the final system for the unknown coefficients in c, we introduce the
m × m bordered matrix
Bb ≡ [ a−N −1 | Ans | aN +1 ] ,
(3.51)
43
and the m × m “evaluator matrix”
Eb ≡
·
b−N −1 | I
(0)
Dn
µ
1
φ0
¶
| bN +1
¸
(3.52)
where the complex transformed velocity is recovered from w a = Eb c.
Equation (3.47) can now be expressed compactly as
Ab c ≡
Ã
Bb − i2κ2 Dm
Ã
1
(φ0 )3/2
!
!
Eb c = D m
Ã
1
(φ0 )3/2
!
F.
(3.53)
The solution for wa (zp ) at the nodal points zp = eph /(1 + eph ) is constructed as
(0)
follows. Since S(j, h) ◦ φ(zp ) = δpj , expansion (3.32), evaluated at the nodal points,
is simply
wa (zp ) = c−N −1 B0 (zp ) + cp
1
φ0 (z
p)
+ cN +1 B1 (zp ).
(3.54)
It follows that the solution for w a at the nodal points are the elements of the column
vector
wa = Eb c = Eb A−1
b Dm
Ã
1
(φ0 )3/2
!
F.
(3.55)
This leads to a solution for the column vectors R(w a ) and I(w a ), which are related
to the approximate nondimensional current components evaluated at the nodal points
by
Ua (zp ) = [R(w a )]p + κ(1 + σ − zp ) cos(χ) = ua (zp ) + κ(1 + σ − zp ) cos(χ)
(3.56)
and
Va (zp ) = [I(w a )]p + κ(1 + σ − zp ) sin(χ) = va (zp ) + κ(1 + σ − zp ) sin(χ).
(3.57)
44
Note that, if so desired, current components may be computed at evenly-spaced grid
points using (3.32) with the coefficients obtained from the solution of (3.53).
Recalling the expansion in (3.32), the velocity can also be written as
Wa (z) = c−N −1 B0 (z) +
N
X
j=−N
cj
S(j, h) ◦ φ(z)
+ cN +1 B1 (z)
φ0 (z)
+ κ(1 + σ − z) (cos(χ) + i sin(χ)) ,
(3.58)
where Ua (z) and Va (z) are the real and imaginary parts of Wa (z).
Numerical Testing: Complex Velocity Formulation
The determinations of the accuracy of the Sinc-Galerkin method when applied to
the constant eddy viscosity case is described next. These results are found in [25].
These simulations, and the illustrative examples which follow later, are carried out
using parameters similar to those that have been used in earlier studies.
Since the governing equations and variables in subsequent developments were
nondimensionalized, the only operative constants in (3.26)-(3.29) are κ, σ, and χ.
In relating these parameters to the “constants of nature”, we adopt the following
nominal values: f = 0.0001 s−1 (appropriate to temperate northern latitudes), sea
water density ρ = 1 × 103 kg m−3 , and air density ρair = 1.25 kg m−3 . Surface wind
stress is assumed to be related to the square of the wind speed Ww (in m s−1 ) by
τw = CD ρair Ww2 ,
(3.59)
45
where the dimensionless parameter CD ≈ 0.0012 for Ww < 12 m s−1 , thereafter
increasing linearly to about 0.0025 at gale force winds (Ww ≈ 30 m s−1 ) [10]. In
keeping with [7] and [9], the linear slip bottom stress coefficient kf is assigned a value
of 0.002 m s−1 for comparison with other work and to dramatize the change in current
speed over the water column. In practice, a value of kf lower by an order of magnitude
may be preferred. Field evidence suggests that the near-surface value of the vertical
eddy viscosity is related to the wind speed. Carter [5] has suggested that, if the wind
is not fetch-limited and the sea state is fully developed, then A∗v (0) in units of m2 s−1
is given by
A∗v (0) ≈ 0.304 × 10−4 Ww3 .
(3.60)
Table 1 shows values of CD , the wind stress τw (N m−2 ) where N represents Newtons,
the kinematic surface eddy viscosity A∗v (0) (m2 s−1 ), the Ekman depth DE (m), and
the seabed depth D0 (m) (when κ = 10) corresponding to wind speeds ranging from
10 m s−1 to 25 m s−1 .
With the parameters and relationships above, and keeping in mind our desire to
compare results with those in [16], we choose our constant eddy viscosity to be
A∗v (z ∗ ) ≡ 0.02 m2 s−1
with τw =
(3.61)
√
2/10 = 0.1414 N m−2 . From Table 1 the wind speed Ww may be
inferred to be about 9 m s−1 . Since DE =
p
2A∗v (0)/f = 20 m, we then have
κ = D0 /DE = D0 /20. In keeping with [16], we will use D0 = 100 m and hence κ = 5,
which it will be throughout.
46
Wind Ww
CD
τw (N m−2 )
A∗v (0) (m2 s−1 )
DE (m)
D0 (m)
10.0
12.0
0.0012
0.15
0.0304
24.7
247.0
0.0012
0.22
0.0526
32.4
324.0
14.0
16.0
18.0
20.0
25.0
0.00134 0.00149 0.00163 0.00178 0.00214
0.33
0.48
0.66
0.89
1.67
0.0835 0.1246 0.1775 0.2434 0.4755
40.9
49.9
59.6
69.8
97.5
409.0
499.0
596.0
698.0
975.0
Table 1. Parameter values corresponding to a range of wind speeds for oceanography
problem.
The numerical results will be compared to the exact solution W ∗ (z ∗ ) = U0 [U (z)+
iV (z)] where U (z) is given by
U (z) = R(Wc (z)) cos(χ) − I(Wc (z)) sin(χ)
and V (z) is given by
V (z) = R(Wc (z)) sin(χ) + I(Wc (z)) cos(χ).
Here R(Wc (z)) and I(Wc (z)) denote the real and imaginary parts of Wc (z), respectively, and
Wc (z) =
κ(1 − i)σcosh (κ(1 − i)(1 − z)) + sinh (κ(1 − i)(1 − z))
.
(1 − i)[cosh (κ(1 − i)) + κ(1 − i)σsinh (κ(1 − i))]
The results of the Sinc-Galerkin approximations Ua (zj ) and Va (zj ) were compared
√
with the exact solutions for U (zj ) and V (zj ) at the sinc grid points with h = π/ 2N
©
ª
S = zj = φ−1 (jh) = ejh /(ejh + 1) : j = −N − 1, . . . , N + 1 .
(3.62)
47
These results were then multiplied by the natural velocity scale U0 to give a dimensional representation of the velocities.
All numerical simulations were run on a SUN BLADE 1000 with MATLAB Version 6.1. To illustrate the performance of the method, the maximum absolute errors
are reported as
kUS k =
−N −1≤j≤N +1
max
{U0 |Ua (zj ) − U (zj )|} ,
kVS k =
−N −1≤j≤N +1
max
{U0 |Va (zj ) − V (zj )|} ,
and
kES k = max {kUS k, kVS k} ,
(3.63)
where the units are m s−1 .
Throughout, comparable graphs (of eddy viscosity functions and velocity components) are shown on the same scale. The horizontal projections of the Ekman spirals
are also shown on the same scale. This way visual comparisons of these various
quantities are readily made.
Example 1. For this example (constant eddy viscosity) we choose χ = 45o and for
the linear stress condition at the seabed we have σ = A∗v (D0 )/(kf D0 ) = 0.1. The
value of N then corresponds to a discrete system of size m × m (m = 2N + 3), given
in (3.53). The errors are given in Table 2 and show a high degree of accuracy.
Figure 11 graphically depicts the numerical convergence of the Sinc-Galerkin method
to the exact solution with the linear stress bottom boundary condition, as N is
48
N
4
8
16
32
PSfrag replacements 64
128
kUS k (m s−1 ) kVS k (m s−1 ) kES k (m s−1 )
m
h
11
19
35
67
131
259
1.111
0.785
0.555
0.393
0.278
0.196
1.080e-03
2.496e-04
2.762e-05
8.985e-07
5.775e-09
4.073e-12
7.478e-04
1.293e-04
1.302e-05
4.240e-07
2.770e-09
1.974e-12
1.080e-03
2.496e-04
2.762e-05
8.985e-07
5.775e-09
4.073e-12
-0.02
Table 2. Errors for Example 1 (constant eddy viscosity) on the sinc grid S with the
linear stress bottom condition for σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m.
Sinc-Galerkin Ekman spiral projections for increasing N
N =4
N =8
N = 16
N = 32
N = 64
True
Northward current component U ∗ (m/s)
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 11. Sinc-Galerkin Ekman spiral projections for Example 1 with increasing
N for constant eddy viscosity with linear stress bottom boundary (σ = 0.1, χ =
45o , κ = 5, D0 = 100 m, DE = 20 m).
49
repeatedly doubled in size. The horizontal projection of the Ekman spiral for N = 64
is indistinguishable from the true solution shown as the solid line.
Example 2. In this example we replace the linear stress bottom boundary condition
with a no-slip condition at the seabed to test the dependence of this scheme on the
parameter σ. This is accomplished by setting σ = 0 in the original formulation. All
other parameters remain the same. The same discrete system given in (3.53) is used
and again the value of N corresponds to a discrete system of size m×m (m = 2N +3).
The errors are given in Table 3 for the same oceanographic conditions and show
similar, remarkably accurate results. For a depth of 100 m, the surface projections
of the Ekman spirals are very nearly the same and for N = 64 are indistinguishable
from the true solution. This is shown in Figure 12.
N
m
h
4
8
16
32
64
128
11
19
35
67
131
259
1.111
0.785
0.555
0.393
0.278
0.196
kUS k (m s−1 ) kVS k (m s−1 ) kES k (m s−1 )
1.071e-03
2.483e-04
2.752e-05
8.957e-07
5.756e-09
4.065e-12
7.500e-04
1.297e-04
1.306e-05
4.255e-07
2.781e-09
1.978e-12
1.071e-03
2.483e-04
2.752e-05
8.957e-07
5.756e-09
4.065e-12
Table 3. Errors for Example 2 on the sinc grid S with the zero-velocity bottom
condition (no-slip) for σ = 0, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m.
50
Sinc-Galerkin Ekman spiral projections for increasing N
N =4
N =8
N = 16
N = 32
N = 64
True
Northward current component U ∗ (m/s)
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 12. Sinc-Galerkin Ekman spiral projections for Example 2 with increasing
N for constant eddy viscosity with no-slip bottom boundary (σ = 0, χ = 45o , κ =
5, D0 = 100 m, DE = 20 m).
Numerical Testing: Variable Eddy Viscosity in Complex Velocity Formulation
In seas of shallow to intermediate depth, at low wind speeds, various factors
such as vigorous tidal mixing are expected to lead to maximum values of A∗v (z ∗ ) at
intermediate depths and minimal values near the surface and seabed. On the other
hand, over deeper water, turbulence generated by high winds produces relatively large
values of A∗v (z ∗ ) near the surface. To illustrate the latter we choose an eddy viscosity
which decreases quadratically from the value of A∗v (0) = 0.02 m2 s−1 to the minimum
51
value of A∗v (D0 ) = 0.00125 m2 s−1 which is given by
PSfrag replacements
A∗v (z ∗ ) = 0.02[1 − (.0075)z ∗ ]2 , 0 < z ∗ < D0 = 100 m .
(3.64)
A graph of the A∗v (z ∗ ) is shown in Figure 13 where it is contrasted with a constant
eddy viscosity of A∗v (z ∗ ) ≡ 0.02 m2 s−1 (from the previous section).
Decreasing eddy viscosity function
0
10
∗
Decreasing A∗
)
v (z
∗
Constant A∗
v (z )
20
Depth z ∗ (m)
30
40
50
60
70
80
90
100
0
0.01
0.02
0.03
0.04
0.05
Eddy viscosity function
0.06
A∗v (z ∗ )
0.07
0.08
0.09
0.1
(m2 /s)
Figure 13. Eddy viscosity functions A∗v (z ∗ ) = 0.02(1 − .0075z ∗ )2 (m2 /s) and A∗v (z ∗ ) ≡
0.02 (m2 /s).
For the purpose of illustrating the application of the Sinc-Galerkin technique to
determine the associated subsurface current distributions, we use parameters from
[16] which include a sea of D0 = 100 m depth, a steady breeze of wind stress τw =
0.14140 N m−2 , and a surface value of the eddy viscosity of A∗v (0) = 0.02 m2 s−1 .
Then from Table 1 the wind speed Ww is inferred to be about 9 m s−1 . Since DE =
52
p
2A∗v (0)/f = 20 m, we then have κ = D0 /DE = 5. The system solved is of size
m × m (m = 2N + 3) and is given in (3.53). The results of the Sinc-Galerkin
approximations Ua (zj ) and Va (zj ) will be illustrated by the Ekman spiral projection
of a complex velocity, the graph of the velocity components, and the Ekman spiral
projection convergence. No closed form solution is available for a direct comparison.
Example 3. For this example in a sea of depth D0 = 100 m, the parameters are
chosen to be σ = 0.1, χ = 45o , and (since DE = 20 m) κ = 5. The decreasing
eddy viscosity function A∗v (z ∗ ) is given by (3.64). Figure 14 shows the Ekman spiral
projection for the decreasing eddy viscosity calculated with N = 128 in comparison
to the same quantity for the constant eddy viscosity. Figure 15 shows the associated
depth variations of the velocity components calculated with N = 128. The same
quantities for the constant eddy viscosity are given for comparison. Ekman spiral
projections for increasing N are superimposed in Figure 16, which again conveys the
rapid convergence of the sinc method.
Example 4. To illustrate the case where the eddy viscosity is minimal near the surface
and seabed the parameters are again chosen to be σ = 0.1, χ = 45o , and (since
DE = 20 m) κ = 5. The quadratic eddy viscosity function is then given by
A∗v (z ∗ ) = 0.02[1 + (.12)z ∗ (1 − (.01)z ∗ )] , 0 < z ∗ < D0 = 100 m
(3.65)
which is shown in Figure 17. Figure 18 shows the Ekman spiral projections for the
PSfrag replacements
53
-0.02
Sinc-Galerkin Ekman spiral projections
0.02
0.02
Northward current component U ∗ (m/s)
∗
Ekman spiral projection for decreasing A∗
)
v (z
∗
Ekman spiral projection for constant A∗
v (z )
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Eastward current component V
∗
0.1
0.12
(m/s)
Figure 14. Sinc-Galerkin Ekman spiral projections for Example 3 with both constant
and decreasing eddy viscosity functions and linear stress bottom boundary (σ =
0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
quadratic eddy viscosity calculated with N = 128 in comparison to the same quantities for the constant eddy viscosity. Figure 19 shows the associated depth variations
of the velocity components calculated with N = 128; the same quantities for the
constant eddy viscosity are given for comparison.
Ekman spiral projections for in-
creasing N are superimposed in Figure 20 which again conveys the rapid convergence
of the sinc method.
54
Sinc-Galerkin velocity components
0
10
20
Depth z ∗ (m)
30
40
50
60
70
PSfrag replacements
∗
U ∗ for decreasing A∗
v (z∗ )
V ∗ for decreasing A∗
(z )
∗
∗v ∗
U for constant Av (z )
∗
V ∗ for constant A∗
v (z )
80
-0.02
90
100
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Velocity components U ∗ and V ∗ (m/s)
Figure 15. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 3 with constant and decreasing eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
Northward current component U ∗ (m/s)
Sinc-Galerkin Ekman spiral projections for increasing N
N =4
N =8
N = 16
N = 32
N = 64
True
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 16. Sinc-Galerkin Ekman spiral projections for Example 3 with increasing
N for the case of the decreasing eddy viscosity function with linear stress bottom
boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
55
Quadratic eddy viscosity function
0
∗
Quadratic A∗
)
v (z
∗
Constant A∗
v (z )
10
20
Depth z ∗ (m)
30
40
50
60
70
80
90
100
0
0.01
0.02
0.03
0.04
0.05
0.06
Eddy viscosity function
A∗v (z ∗ )
0.07
0.08
0.09
0.1
(m2 /s)
PSfrag replacements
Figure 17. Eddy viscosity functions A∗v (z ∗ ) = 0.02[1 + (.12)z ∗ (1 − (.01)z ∗ )] and
-0.02
A∗v (z ∗ ) ≡ 0.02 (m2 /s).
0.02
Sinc-Galerkin Ekman spiral projections
Northward current component U ∗ (m/s)
0.02
∗
Ekman spiral projection for quadratic A∗
v (z∗ )
Ekman spiral projection for constant A∗
v (z )
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 18. Sinc-Galerkin Ekman spiral projections for Example 4 with both constant and quadratic eddy viscosity functions and linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
56
Sinc-Galerkin velocity components
0
10
20
Depth z ∗ (m)
30
40
50
60
70
∗
U ∗ for quadratic A∗
v (z∗ )
V ∗ for quadratic A∗
v (z∗ )
∗
∗
U for constant Av (z )
∗
V ∗ for constant A∗
v (z )
80
90
100
-0.1
-0.08
-0.06
-0.04
-0.02
0
Velocity components
0.02
U∗
and V
0.04
∗
0.06
0.08
0.1
(m/s)
Figure 19. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 4 with constant and quadratic eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
Sinc-Galerkin Solution of the Coupled Differential Equation System
In the previous section, the equations (3.26)-(3.29) are solved with the SincGalerkin procedure. This is the complex velocity formulation. In this section, we are
interested in solving directly the coupled differential equation system in (3.21)-(3.24)
which is defined on the same domain. This is a new procedure which was not used
in [25]. The two procedures will be shown to give nearly identical results.
The coupled differential equations (3.21)-(3.24) are given by
Lu(z) + 2κ2 v(z) = F1 (z)
Lv(z) − 2κ2 u(z) = F2 (z)
(3.66)
57
Sinc-Galerkin Ekman spiral projections for increasing N
Northward current component U ∗ (m/s)
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
N =4
N =8
N = 16
N = 32
N = 64
True
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 20. Sinc-Galerkin Ekman spiral projections for Example 4 with increasing
N for the case of the quadratic eddy viscosity function with linear stress bottom
boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
where
d
Lu(z) ≡ −
dz
µ
du(z)
Av (z)
dz
¶
,
d
Lv(z) ≡ −
dz
µ
dv(z)
Av (z)
dz
¶
0
F1 (z) = −2κ3 (1 + σ − z) sin(χ) − κ cos(χ)Av (z)
0
F2 (z) = 2κ3 (1 + σ − z) cos(χ) − κ sin(χ)Av (z).
(3.67)
The surface conditions are
du(0)
=0 ,
dz
dv(0)
= 0,
dz
(3.68)
while the seabed conditions are
u(1) + σ
du(1)
dv(1)
= 0 , v(1) + σ
= 0.
dz
dz
(3.69)
58
For the Sinc-Galerkin method, the approximate solutions to u(z) and v(z), subject
to the mixed conditions (3.68) and (3.69) are represented by the expansions
ua (z) = c−N −1 B0 (z) +
va (z) = d−N −1 B0 (z) +
N
X
cj
S(j, h) ◦ φ(z)
+ cN +1 B1 (z),
φ0 (z)
dj
S(j, h) ◦ φ(z)
+ dN +1 B1 (z),
φ0 (z)
j=−N
N
X
j=−N
(3.70)
where B0 (z) and B1 (z) are defined in (3.31).
Applying the same Sinc-Galerkin procedure to both equations in (3.66), the result
will be linear equations whose solutions give the coefficients of the expansions in (3.70)
for ua (z) and va (z) which is similar to (3.47). For p = −N −1, −N, . . . , 0, . . . , N, N +1,
¾
N ½
X
−1 (2) 1 (0)
LB0 (zp )
−1/2
cj
c−N −1 +
δ + δpj Av (zj ) (φ0 (zj ))
2 pj
(φ0 (zp ))3/2
h
4
j=−N
¾
½
N
X −1 (1) 1
−3/2
(0)
+
δpj + (2zj − 1) δpj A0v (zj ) (φ0 (zj ))
cj
h
2
j=−N
+
(3.71)
LB1 (zp )
F1 (zp )
cN +1 + 2κ2 (φ0 (zp ))−3/2 va (zp ) = 0
,
0
3/2
(φ (zp ))
(φ (zp ))3/2
and
¾
N ½
X
−1 (2) 1 (0)
LB0 (zp )
−1/2
dj
d−N −1 +
δ + δpj Av (zj ) (φ0 (zj ))
2 pj
(φ0 (zp ))3/2
h
4
j=−N
½
¾
N
X −1 (1) 1
−3/2
(0)
+
dj
δpj + (2zj − 1) δpj A0v (zj ) (φ0 (zj ))
h
2
j=−N
+
(3.72)
LB1 (zp )
F2 (zp )
dN +1 − 2κ2 (φ0 (zp ))−3/2 ua (zp ) = 0
.
0
3/2
(φ (zp ))
(φ (zp ))3/2
Introduce the column vectors c and d that represent the coefficients of the approximate solutions in (3.70),
59
c ≡ [c−N −1 c−N · · · cN cN +1 ]T ,
d ≡ [d−N −1 d−N · · · dN dN +1 ]T ,
and define
F 1 ≡ [F1 (z−N −1 ) F1 (z−N ) · · · F1 (zN ) F1 (zN +1 )]T ,
F 2 ≡ [F2 (z−N −1 ) F2 (z−N ) · · · F2 (zN ) F2 (zN +1 )]T ,
ua ≡ [ua (z−N −1 ) ua (z−N ) · · · ua (zN ) ua (zN +1 )]T ,
v a ≡ [va (z−N −1 ) va (z−N ) · · · va (zN ) va (zN +1 )]T ,
where F1 (z) and F2 (z) are given in (3.67). From (3.71) and (3.72) we arrive at the
discrete system
µ
¶
¶
1
1
Eb d = D m
F1
Bb c + 2κ Dm
(φ0 )3/2
(φ0 )3/2
¶
¶
µ
µ
1
1
2
Bb d − 2κ Dm
Eb c = D m
F 2,
(φ0 )3/2
(φ0 )3/2
2
µ
where Bb and Eb are defined in (3.51) and (3.52), respectively.
This leads to the coupled linear discrete system
AX = C
where the (2m) × (2m) block matrix (m = 2N + 3)
´ 
³

1
2
Bb
2κ Dm (φ0 )3/2 Eb






A = 



´
³
1
2
−2κ Dm (φ0 )3/2 Eb
Bb
(3.73)
60
and the (2m) × 1 column vectors


X = 

c
d




C=



,

Dm
³
Dm
³
1
0
(φ )3/2
1
0
(φ )3/2
´
F1
´
F2



.


The solutions for ua (zp ) and va (zp ) at the nodal points zp = eph /(1 + eph ) are con(0)
structed as follows. Since S(j, h) ◦ φ(zp ) = δpj , expansion (3.70), evaluated at the
nodal points, is simply
ua (zp ) = c−N −1 B0 (zp ) + cp
1
φ0 (z
va (zp ) = d−N −1 B0 (zp ) + dp
p)
1
φ0 (z
p)
+ cN +1 B1 (zp ),
+ dN +1 B1 (zp ).
It follows that the solutions for ua and v a at the nodal points are the elements of this
column vector




ua
va


 
=
 
Eb
O
O
Eb




where the (2m) × (2m) block evaluator matrix is


O
Eb






O
Eb
c
d


,

(3.74)
and O is an m × m zero matrix. These solutions are related to the approximate
nondimensional current components evaluated at the nodal points in (3.14) as
Ua (zp ) = ua (zp ) + κ(1 + σ − zp ) cos(χ) and Va (zp ) = va (zp ) + κ(1 + σ − zp ) sin(χ).
61
Numerical Testing: Coupled System
Example 5. For the same conditions as in Example 1, we find approximate solutions Ua (z) and Va (z), respectively, using the coupled discrete system in (3.73) where
U0 Ua (z) = Ua∗ (z ∗ ) and U0 Va (z) = Va∗ (z ∗ ). The errors are shown in Table 4. The
horizontal projection of the Ekman spirals is shown in Figure 21.
N
2m
h
4
8
16
32
64
128
22
38
70
134
262
518
1.111
0.785
0.555
0.393
0.278
0.196
kUS k (m s−1 ) kVS k (m s−1 ) kES k (m s−1 )
1.080e-03
2.496e-04
2.762e-05
8.985e-07
5.775e-09
4.053e-12
7.478e-04
1.293e-04
1.302e-05
4.240e-07
2.770e-09
1.937e-12
1.080e-03
2.496e-04
2.762e-05
8.985e-07
5.775e-09
4.053e-12
Table 4. Errors for Example 5 (constant eddy viscosity) on the sinc grid S with the
linear stress bottom condition for σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m.
Example 6. From Example 2 (no-slip condition at the seabed) we set σ = 0 and
solve for approximate solutions Ua (z) and Va (z), respectively, by using the coupled
discrete system in (3.73). The reported errors are shown in Table 5 and the horizontal
projection of the Ekman spirals is shown in Figure 22.
62
Sinc-Galerkin Ekman spiral projections for increasing N
N =4
N =8
N = 16
N = 32
N = 64
True
Northward current component U ∗ (m/s)
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 21. Sinc-Galerkin Ekman spiral projections for Example 5 with increasing N
for the case of constant eddy viscosity, A∗v (z ∗ ) ≡ 1, with linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
N
2m
h
4
8
16
32
64
128
22
38
70
134
262
518
1.111
0.785
0.555
0.393
0.278
0.196
kUS k (m s−1 ) kVS k (m s−1 ) kES k (m s−1 )
1.071e-03
2.483e-04
2.752e-05
8.957e-07
5.756e-09
4.073e-12
7.500e-04
1.297e-04
1.306e-05
4.255e-07
2.781e-09
1.975e-12
1.071e-03
2.483e-04
2.752e-05
8.957e-07
5.756e-09
4.073e-12
Table 5. Errors for Example 6 (constant eddy viscosity) on the sinc grid S with the
zero-velocity bottom condition for σ = 0, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m.
63
Sinc-Galerkin Ekman spiral projections for increasing N
N =4
N =8
N = 16
N = 32
N = 64
True
Northward current component U ∗ (m/s)
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 22. Sinc-Galerkin Ekman spiral projections for Example 6 with increasing N
for the case of constant eddy viscosity, A∗v (z ∗ ) ≡ 1, with no-slip bottom boundary
(σ = 0, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
Numerical Testing: The variable Eddy Viscosity in Coupled System Formulation
The following examples show the results for the equations for the decreasing and
quadratic eddy viscosity functions given by (3.64) and (3.65). They are solved by the
coupled discrete system in (3.73) of size a 2m × 2m (m = 2N + 3). Also the results
of the complex velocity system and the coupled system will be compared in Table 6
and Table 7.
Example 7. From Example 3 the decreasing eddy viscosity function A∗v (z ∗ ) is given
by (3.64). We find the approximate solutions Ua (z) and Va (z) by using the coupled
64
discrete system in (3.73). The results are shown in Figure 23, Figure 24, and Figure
25.
PSfrag replacements
-0.02
Sinc-Galerkin Ekman spiral projections
Northward current component U ∗ (m/s)
0.02
0.02
∗
Ekman spiral projection for quadratic A∗
v (z∗ )
Ekman spiral projection for constant A∗
v (z )
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Eastward current component V ∗ (m/s)
Figure 23. Sinc-Galerkin Ekman spiral projections for Example 7 with both constant
and decreasing eddy viscosity functions and linear stress bottom boundary (σ =
0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
Example 8. From the same conditions as in Example 4 with the quadratic eddy
viscosity function in (3.65), the approximate solutions Ua (z) and Va (z) are illustrated
by graphs in Figure 26, Figure 27, and Figure 28. The approximations are calculated
from the 2m × 2m coupled discrete system in (3.73).
65
Sinc-Galerkin velocity components
0
10
20
Depth z ∗ (m)
30
40
50
60
70
PSfrag replacements
∗
U ∗ for decreasing A∗
)
v (z
V ∗ for decreasing A∗
(z ∗ )
∗
∗v ∗
U for constant Av (z )
∗
V ∗ for constant A∗
v (z )
80
-0.02
90
100
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Velocity components U ∗ and V ∗ (m/s)
Figure 24. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 7 with constant and decreasing eddy viscosity functions and linear stress
bottom boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m) .
Northward current component U ∗ (m/s)
Sinc-Galerkin Ekman spiral projections for increasing N
N =4
N =8
N = 16
N = 32
N = 64
True
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Eastward current component V ∗ (m/s)
Figure 25. Sinc-Galerkin Ekman spiral projections for Example 7 with increasing
N for the case of the decreasing eddy viscosity function and linear stress bottom
boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
PSfrag replacements
-0.02
66
0.02
Sinc-Galerkin Ekman spiral projections
Northward current component U ∗ (m/s)
0.02
∗
Ekman spiral projection for quadratic A∗
v (z∗ )
Ekman spiral projection for constant A∗
v (z )
0.01
0
-0.01
PSfrag replacements
-0.02
-0.03
-0.04
-0.02
0.02
0
0.04
0.06
0.08
Eastward current component V
∗
0.1
0.12
(m/s)
Figure 26. Sinc-Galerkin Ekman spiral projections for Example 8 with both constant and quadratic eddy viscosity functions and linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
Sinc-Galerkin velocity components
0
10
20
Depth z ∗ (m)
30
40
50
60
70
∗
U ∗ for quadratic A∗
)
v (z
∗
V ∗ for quadratic A∗
v (z∗ )
∗
∗
U for constant Av (z )
∗
V ∗ for constant A∗
v (z )
80
90
100
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Velocity components U ∗ and V ∗ (m/s)
Figure 27. Sinc-Galerkin northward and eastward calculated velocity profiles for
Example 8 with constant and quadratic eddy viscosity functions and linear stress
bottom boundary(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m) .
67
Sinc-Galerkin Ekman spiral projections for increasing N
northward current component U ∗ (m/s)
0.01
0.005
0
−0.005
-0.01
-0.015
-0.02
N =4
N =8
N = 16
N = 32
N = 64
True
-0.025
-0.03
-0.035
-0.04
-0.02
0
0.02
0.04
0.06
Eastward current component V
0.08
∗
0.1
0.12
(m/s)
Figure 28. Sinc-Galerkin Ekman spiral projections for Example 8 with increasing N
for the case of the quadratic eddy viscosity function and linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
Next, we want to compare the approximate solutions Ua (z) and Va (z) for the
complex velocity system and the coupled system. To illustrate the performance of
both methods, the maximum absolute errors are reported as
kUC k =
−N −1≤j≤N +1
max
{U0 |U1 (zj ) − U2 (zj )|} ,
kVC k =
−N −1≤j≤N +1
max
{U0 |V1 (zj ) − V2 (zj )|} ,
kEC k = max {kUC k, kVC k} ,
where U1 (z) and V1 (z) are computed by the complex velocity system in (3.53) and
U2 (z) and V2 (z) are computed by the coupled system in (3.73).
68
N
m
4
8
16
32
64
128
11
19
35
67
131
259
2m
kUC k
22 1.005e-15
38 9.419e-16
70 6.279e-16
134 1.130e-15
262 5.651e-15
518 1.463e-14
kVC k
kEC k
8.163e-16 1.005e-15
1.193e-15 1.193e-15
6.217e-15 6.217e-15
2.009e-15 2.009e-15
1.155e-14 1.155e-14
1.758e-14 1.758e-14
Table 6. Comparison between the approximate solutions for the complex velocity
system and the coupled system by using the same sinc grid size for Examples 1
and 5 for the case of constant eddy viscosity with linear stress bottom boundary
(σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
N
m
4
8
16
32
64
128
11
19
35
67
131
259
2m
kUC k
22 1.884e-16
38 3.768e-16
70 2.198e-16
134 1.381e-15
262 4.969e-15
518 4.458e-14
kVC k
kEC k
9.419e-16 9.419e-16
1.193e-15 1.193e-15
3.579e-15 3.579e-15
2.261e-15 2.261e-15
1.068e-14 1.068e-14
2.003e-14 4.458e-14
Table 7. Comparison between the approximate solutions for the complex velocity
system and the coupled system on the same sinc grid for Examples 4 and 8 for the
decreasing eddy viscosity A∗v (z ∗ ) = .02(1 + .12z ∗ (1 − .01z ∗ )) with linear stress bottom
boundary (σ = 0.1, χ = 45o , κ = 5, D0 = 100 m, DE = 20 m).
69
CHAPTER 4
WIND-DRIVEN CURRENTS IN A SEA WITH A TIME-DEPENDENT EDDY
VISCOSITY
In the previous chapter, we studied the wind-driven current velocity in a sea with
an eddy viscosity which varied with the depth of the ocean. Now we are interested
in the current velocity in a sea with an eddy viscosity that is time-dependent. The
current velocity is the solution of a partial differential equation which models ocean
circulation.
Governing Equations
First, we discuss some assumptions made in wind-driven studies. We will then use
the Sinc-Galerkin method to simulate the model equations. The horizontal wind-drift
current velocity is
q ∗ (z ∗ , t∗ ) = U ∗ (z ∗ , t∗ )x̂∗ + V ∗ (z ∗ , t∗ )ŷ ∗ .
This current is driven by a time-dependent tangential surface wind stress of magnitude
τw ψ(t∗ ) at the surface (z ∗ = 0) represented as
¡
¢
τ (0, t∗ ) = τw ψ(t∗ ) cos(χ(t∗ ))x̂∗ + sin(χ(t∗ ))ŷ ∗ ,
where χ(t∗ ) is the angle between the positive x∗ -axis and the wind direction in Figure
(29).
70
x∗
q ∗ (z ∗, t∗ )
Sea surface
χ(t∗ )
y∗
z∗ = 0
τ (0, t∗) = τw ψ(t∗ )(cos(χ(t∗))x̂∗ + sin(χ(t∗ ))ŷ∗)
τ (z ∗ , t∗ ) = −ρA∗v (t∗ )∂q∗/∂z ∗
A∗v (t∗), time-dependent eddy viscosity
z∗
z ∗ = D0 , Seabed
Figure 29. The general physical model of time-dependent eddy viscosity oceanography
problem.
In a specified eddy viscosity model, the time-dependent eddy viscosity coefficient
A∗v (t∗ ) is defined by
¡
¢
A∗v (t∗ ) = A0 1 + κw ψ(t∗ )
which is assumed to be a continuously differentiable function of t∗ ∈ [0, ∞). The
internal frictional stresses are given by
τ (z ∗ , t∗ ) = −ρA∗v (t∗ )
∂q ∗ (z ∗ , t∗ )
,
∂z ∗
where the ocean mass density is ρ. The wind-drift current velocity q ∗ (z ∗ , t∗ ) can be
determined by solving
µ
¶
∗ ∗ ∗
∂q ∗ (z ∗ , t∗ )
∂
∗ ∗ ∂q (z , t )
− ∗ Av (t )
= f ẑ ∗ × q ∗ (z ∗ , t∗ ),
∗
∗
∂t
∂z
∂z
0 < z ∗ < D0 ,
0 < t∗ .
(4.1)
71
The stress condition at the sea surface, z ∗ = 0, is equal to the tangential surface
time-dependent wind stress
−ρA∗v (t∗ )
∂q ∗ (0, t∗ )
= τw ψ(t∗ )(cos(χ(t∗ ))x̂∗ + sin(χ(t∗ ))ŷ ∗ ),
∂z ∗
0 < t∗ .
(4.2)
The frictional stress is assumed linearly proportional to the current at the seabed,
z ∗ = D0 ,
−ρA∗v (t∗ )
∂q ∗ (D0 , t∗ )
= kf ρq ∗ (D0 , t∗ ),
∗
∂z
0 < t∗ .
(4.3)
Initially the sea is assumed to be at rest, so
q ∗ (z ∗ , 0) = 0,
0 < z ∗ < D0 .
(4.4)
Here f ≡ 2Ω sin(θ) is the Coriollis parameter at latitude θ, where Ω = 7.29 × 10 −5
rad s−1 is the angular speed of rotation of the earth and κf is the linear slip bottom
stress coefficient.
From (4.1), we can simplify the initial-boundary-value problem
µ
¶
∗ ∗ ∗
∂q ∗ (z ∗ , t∗ )
∂
∗ ∗ ∂q (z , t )
− ∗ Av (t )
∂t∗
∂z
∂z ∗
· ∗ ∗ ∗
µ
¶¸
∗ ∗ ∗
∂U (z , t )
∂
∗ ∗ ∂U (z , t )
=
− ∗ Av (t )
x̂∗
∂t∗
∂z
∂z ∗
µ
¶¸
· ∗ ∗ ∗
∗ ∗ ∗
∂
∂V (z , t )
∗ ∗ ∂V (z , t )
− ∗ Av (t )
ŷ ∗
+
∂t∗
∂z
∂z ∗
= f ẑ ∗ × q ∗ (z ∗ , t∗ )
= f ẑ ∗ × [U ∗ (z ∗ , t∗ )x̂∗ + V ∗ (z ∗ , t∗ )ŷ ∗ ]
= −f [V ∗ (z ∗ , t∗ )x̂∗ − U ∗ (z ∗ , t∗ )ŷ ∗ ] .
72
Then it can be separated into component form as
µ
¶
∗ ∗ ∗
∂
∂U ∗ (z ∗ , t∗ )
∗ ∗ ∂U (z , t )
= −f V ∗ (z ∗ , t∗ )
− ∗ Av (t )
∂t∗
∂z
∂z ∗
(4.5)
µ
¶
∗ ∗ ∗
∂V ∗ (z ∗ , t∗ )
∂
∗ ∗ ∂V (z , t )
− ∗ Av (t )
= f U ∗ (z ∗ , t∗ ).
∗
∗
∂t
∂z
∂z
(4.6)
and
The boundary conditions for (4.5) at the sea surface, z ∗ = 0, and the seabed, z ∗ = D0 ,
are
−ρA∗v (t∗ )
∂U ∗ (0, t∗ )
= τw ψ(t∗ ) cos(χ(t∗ )),
∂z ∗
0 < t∗
(4.7)
and
−ρA∗v (t∗ )
∂U ∗ (D0 , t∗ )
= kf ρU ∗ (D0 , t∗ ),
∂z ∗
0 < t∗ .
(4.8)
The initial condition is
U ∗ (z ∗ , 0) = 0,
0 < z ∗ < D0 .
(4.9)
The boundary conditions for (4.6) at the sea surface, z ∗ = 0, and the seabed, z ∗ = D0 ,
are
−ρA∗v (t∗ )
∂V ∗ (0, t∗ )
= τw ψ(t∗ ) sin(χ(t∗ )),
∂z ∗
0 < t∗
(4.10)
0 < t∗ .
(4.11)
and
−ρA∗v (t∗ )
∂V ∗ (D0 , t∗ )
= kf ρV ∗ (D0 , t∗ ),
∂z ∗
73
The initial condition is
V ∗ (z ∗ , 0) = 0,
0 < z ∗ < D0 .
(4.12)
To nondimensionalize the initial-boundary-value problem (4.5)-(4.12), define the nondimensional variables
DE ≡
s
√
2A0
2τw
τ w DE
, U0 ≡
= √
,
f
ρA0
ρ A0 f
t∗
A∗ (t∗ )
, A0 ≡ A∗v (0),
t≡
, Av (t) ≡ v∗
T0
Av (0)
r
f
A0
D0
= D0
, σ≡
, fc ≡ f T 0
κ≡
DE
2A0
κ f D0
q ∗ (z ∗ , t∗ )
U ∗ (z ∗ , t∗ )
V ∗ (z ∗ , t∗ )
q(z, t) ≡
, U (z, t) ≡
, V (z, t) ≡
, (4.13)
U0
U0
U0
z≡
z∗
,
D0
where T0 is the fundamental time scale.
We apply (4.13) to nondimensionalize the partial differential equations (4.5) and
(4.6). For (4.5), we get
µ
¶
1 ∂(U0 U (z, t)) A0 ∂
∂(U0 U (z, t))
Av (t)
= −f U0 V (z, t),
− 2
T0
∂t
D0 ∂z
∂z
which results in
µ
¶
∂U (z, t)
fc ∂
∂U (z, t)
− 2
Av (t)
= −fc V (z, t),
∂t
2κ ∂z
∂z
where
0 < z < 1,
0 < t.
For (4.6), we get
µ
¶
1 ∂(U0 V (z, t)) A0 ∂
∂(U0 V (z, t))
− 2
Av (t)
= f U0 U (z, t),
T0
∂t
D0 ∂z
∂z
(4.14)
74
which results in
µ
¶
∂V (z, t)
fc ∂
∂V (z, t)
Av (t)
= fc U (z, t),
− 2
∂t
2κ ∂z
∂z
(4.15)
where
0 < z < 1,
0 < t.
From (4.7) and (4.10), the surface boundary conditions are nondimensionalized. For
(4.7), we have
−ρ
A0 Av (t) ∂(U0 U (0, t))
= τw ψ(t) cos(χ(t)),
D0
∂z
0 < t,
which leads to
−ρ
A0 Av (t) τw DE ∂U (0, t)
= τw ψ(t) cos(χ(t)),
D0
ρA0
∂z
0 < t,
or
∂U (0, t)
−κψ(t) cos(χ(t))
=
,
∂z
Av (t)
0 < t.
(4.16)
For (4.10), we have
−ρ
A0 Av (t) ∂(U0 V (0, t))
= τw ψ(t) sin(χ(t)),
D0
∂z
0 < t,
which leads to
−ρ
A0 Av (t) τw DE ∂V (0, t)
= τw ψ(t) sin(χ(t)),
D0
ρA0
∂z
0 < t,
or
−κψ(t) sin(χ(t))
∂V (0, t)
=
,
∂z
Av (t)
0 < t.
(4.17)
75
The seabed boundary conditions from (4.8) and (4.11) are also nondimensionalized.
For (4.8), we have
−
A0 Av (t) ∂(U0 U (1, t))
= kf U0 U (1, t),
D0
∂z
0 < t,
which gives
U (1, t) + σAv (t)
∂U (1, t)
= 0,
∂z
0 < t.
(4.18)
For (4.11), we have
−
A0 Av (t) ∂(U0 V (1, t))
= kf U0 V (1, t),
D0
∂z
0 < t,
which gives
V (1, t) + σAv (t)
∂V (1, t)
= 0.
∂z
(4.19)
The initial conditions in (4.9) and (4.12) become
U (z, 0) = 0,
V (z, 0) = 0,
0 < z < 1.
(4.20)
From (4.14)-(4.20), the nondimensional initial-boundary -value problem that governs the time-dependent velocity can be written in component form as
µ
¶
fc ∂
∂U (z, t)
∂U (z, t)
− 2
Av (t)
∂t
2κ ∂z
∂z
=
−fc V (z, t), 0 < z < 1,
0 < t, (4.21)
fc U (z, t), 0 < z < 1,
0 < t, (4.22)
and
µ
¶
∂V (z, t)
fc ∂
∂V (z, t)
− 2
Av (t)
∂t
2κ ∂z
∂z
=
76
with surface boundary conditions
ψ(t)
∂U (0, t)
= −κ
cos(χ(t)),
∂z
Av (t)
∂V (0, t)
ψ(t)
= −κ
sin(χ(t)),
∂z
Av (t)
0 < t, (4.23)
and seabed boundary conditions
U (1, t) + σAv (t)
∂U (1, t)
= 0,
∂z
V (1, t) + σAv (t)
∂V (1, t)
= 0,
∂z
0 < t.
(4.24)
Lastly, the initial conditions are
U (z, 0) = 0,
V (z, 0) = 0,
0 < z < 1.
(4.25)
We first transform the nonhomogeneous boundary conditions to homogeneous
boundary conditions by using the linear transformations
·
¸
ψ(t)
U (z, t) = u(z, t) + κ(1 − z)
+ κσψ(t) cos(χ(t)),
Av (t)
·
¸
ψ(t)
+ κσψ(t) sin(χ(t)).
V (z, t) = v(z, t) + κ(1 − z)
Av (t)
(4.26)
In the transformation procedure, we must take the partial derivative once with respect
to t, and the partial derivative twice with respect to z, of U and V in (4.26). This
yields
¡
ψ(t)
∂U (z, t)
∂u(z, t) ∂ [κ(1 − z) Av (t) + κσψ(t)] cos(χ(t)))
=
+
,
∂t
∂t
∂t
¡
ψ(t)
∂v(z, t) ∂ [κ(1 − z) Av (t) + κσψ(t)] sin(χ(t)))
∂V (z, t)
=
+
,
∂t
∂t
∂t
∂u(z, t)
ψ(t)
∂U (z, t)
=
−κ
cos(χ(t)),
∂z
∂z
Av (t)
∂v(z, t)
ψ(t)
∂V (z, t)
=
−κ
sin(χ(t)),
∂z
∂z
Av (t)
(4.27)
(4.28)
77
and
µ
¶
µ
¶
∂U (z, t)
∂
∂u(z, t)
∂
Av (t)
=
Av (t)
,
∂z
∂z
∂z
∂z
µ
¶
µ
¶
∂
∂V (z, t)
∂
∂v(z, t)
Av (t)
=
Av (t)
.
∂z
∂z
∂z
∂z
(4.29)
So (4.21) and (4.22) are transformed to
¶
∂u(z, t)
Av (t)
∂z
£
¤
ψ(t)
= −fc v(z, t) − fc κ(1 − z)
+ κσψ(t) sin(χ(t))
Av (t)
ψ(t)
∂¡
[κ(1 − z)
+ κσψ(t)] cos(χ(t)))
−
∂t
Av (t)
∂u(z, t)
fc ∂
− 2
∂t
2κ ∂z
µ
(4.30)
µ
(4.31)
and
¶
∂v(z, t)
Av (t)
∂z
£
¤
ψ(t)
= fc u(z, t) + fc κ(1 − z)
+ κσψ(t) cos(χ(t))
Av (t)
¡
∂
ψ(t)
−
+ κσψ(t)] sin(χ(t))).
[κ(1 − z)
∂t
Av (t)
∂v(z, t)
fc ∂
− 2
∂t
2κ ∂z
Transforming to the complex plane, we multiply (4.31) by the imaginary unit i, add
the result to (4.30) and define the complex velocity w(z, t) = u(z, t) + iv(z, t). Then
the resulting partial differential equation is
∂w(z, t)
fc ∂
− 2
∂t
2κ ∂z
µ
¶
∂w(z, t)
Av (t)
− ifc w(z, t)
∂z
¸
·
ψ(t)
= ifc κ(1 − z)
+ κσψ(t) eiχ(t)
Av (t)
#
"
¶0
µ
0
¡
¢
ψ(t) iχ(t)
+ κσ ψ(t)eiχ(t)
e
− κ(1 − z)
Av (t)
78
¶0
¸
ψ(t) iχ(t)
ψ(t) iχ(t)
e
e
= −κ(1 − z)
− ifc
Av (t)
Av (t)
i
h¡
¢0
iχ(t)
iχ(t)
− ifc ψ(t)e
= κσ ψ(t)e
·µ
= −κ(1 − z)r(t) − κσg(t),
(4.32)
where
0
0
r(t) ≡ Φ (t) − ifc Φ(t),
g(t) ≡ Ψ (t) − ifc Ψ(t),
(4.33)
and
Φ(t) ≡
ψ(t) iχ(t)
e
,
Av (t)
Ψ(t) ≡ ψ(t)eiχ(t) .
If one defines
Hw(z, t)
≡
µ
¶
∂w(z, t)
fc ∂
∂w(z, t)
− 2
Av (t)
∂t
2κ ∂z
∂z
the partial differential equation is now given by
Hw(z, t) − ifc k2 w(z, t)
=
F (z, t), 0 < z < 1, 0 < t
(4.34)
where k2 = 1,
F (z, t)
≡
−κ(1 − z)r(t) − κσg(t),
and the functions r(t) and g(t) are in (4.33). Later we will also choose k2 = 0 in
order to test our numerical scheme. Repeating this procedure with (4.23), the surface
conditions reduce to
∂w(0, t)
∂z
=
0,
0<t
(4.35)
79
and with (4.24), the seabed conditions become
w(1, t) + σAv (t)
∂w(1, t)
∂z
=
0,
0 < t.
(4.36)
The initial conditions in (4.25) are then
w(z, 0) = 0,
0 < z < 1.
(4.37)
Solving the Problem with Time-Independent Boundary Conditions
In this section, the Sinc-Galerkin method is first applied to a simplified version of
(4.34)-(4.37) where fc /2κ2 ≡ 1 and k2 = 0. The time-dependent boundary condition
has been changed to a constant multiple σ so
µ
¶
∂w(z, t)
∂
∂w(z, t)
Av (t)
= F (z, t), 0 < z < 1, 0 < t
−
Lw(z, t) ≡
∂t
∂z
∂z
(4.38)
subject to the time-independent boundary conditions
∂w(0, t)
= 0, 0 < t,
∂z
w(1, t) + σ
∂w(1, t)
= 0, 0 < t,
∂z
(4.39)
(4.40)
with the initial condition
w(z, 0) = 0, 0 < z < 1.
(4.41)
Both temporal and spatial spaces are introduced for the method. The SincGalerkin procedure begins by selecting composite sinc functions appropriate to each
80
domain (0, 1) and (0, ∞) as the sinc basis functions for the approximate expansion
of the complex current velocity w(z, t). The conformal mapping function in (2.1) on
the spatial domain (0, 1) is given by
¶
z
,
φ(z) = ln
1−z
µ
and the appropriate composite sinc functions S(p, hz ) ◦ φ over (0, 1) are given by
´
³

π

(φ(z)
−
ph
)
sin
z
hz

µ
¶ 

, φ(z) 6= phz
φ(z) − phz
π
(φ(z) − phz )
S(p, hz ) ◦ φ(z) ≡ sinc
≡
hz

hz



1
, φ(z) = phz .
The conformal mapping function (2.5) on the temporal domain (0, ∞) is
Υ(t) = ln(t),
and the appropriate composite sinc functions S(q, ht ) ◦ Υ over (0, ∞) are given by
´
³

π

sin ht (Υ(t) − qht )

¶ 
µ

, Υ(t) 6= qht
Υ(t) − qht
π
(Υ(t) − qht )
≡
S(q, ht ) ◦ Υ(t) ≡ sinc
ht

ht



1
, Υ(t) = qht .
Consider the set of sinc basis functions for the two-dimensional problem (4.38)-(4.41)
as the product sinc basis functions
©
¡
¢¡
¢ª
Sp (z)Sq∗ (t) ≡ S(p, hz ) ◦ φ(z) S(q, ht ) ◦ Υ(t)
(4.42)
for −Mz − 1 ≤ p ≤ Nz + 1 and −Mt ≤ q ≤ Nt + 1.
For the boundary conditions in (4.39) and (4.40), the sinc basis functions (4.42)
tend to zero as z approaches to 0 and 1 but their z derivatives are undefined at z = 0
81
and z = 1. To remedy this it is necessary to modify the sinc basis functions and add
some additional boundary basis functions. The Hermite functions, whose derivatives
are defined at the two endpoints, are selected to be boundary basis functions for the
spatial space as
w0 (z) = (2z + 1)(1 − z)2 ,
w1 (z) = (1 − z)z 2 + σ(3 − 2z)z 2 .
(4.43)
Also the sinc basis functions (4.42) tend to zero as t approaches ∞, so a nonzero
bounded function is chosen to be an additional basis function for the temporal space
w∞ (t) =
t
.
1+t
(4.44)
The approximate solution for w(z, t) is represented by the expansion,
(mz = Mz + Nz + 3, nz = Mz + Nz + 1, mt = Mt + Nt + 2, nt = Mt + Nt + 1),
wa (z, t) =
N
z +1
X
N
t +1
X
cjk ξj (z)ζk (t)
(4.45)
j=−Mz −1 k=−Mt
where

w0 (z),






 µS ¶
j
ξj (z) =
(z),

φ0






w1 (z),
 ∗
 Sk (t),
ζk (t) =

w∞ (t),
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1,
if k = −Mt , . . . , Nt
if k = Nt + 1.
The unknown coefficients cjk will ultimately be stored in the mz × mt coefficient
matrix C = [cjk ].
82
We first show that the approximate solution in (4.45) satisfies the boundary conditions (4.39) and (4.40). First we compute
dξj (z)
and evaluate them at the endpoints,
dz
z = 0 and z = 1, as
 0
w0 (0) = 0,





 µ ¶0
dξj (0)  Sj
=
(0) = 0,
0

dz
φ





 0
w1 (0) = 0,
Also
 0
w0 (1) = 0,





 µ ¶0
dξj (1)  Sj
=
(1) = 0,

dz
φ0





 0
w1 (1) = −1,

w0 (0) = 1,






 µS ¶
j
ξj (0) =
(0) = 0,
0

φ






w1 (0) = 0,

w0 (1) = 0,






 µS ¶
j
ξj (1) =
(1) = 0,
0

φ






w1 (1) = σ,
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1 ,
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1.
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1,
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1.
Next we evaluate ζk (t) at t = 0, and t = ∞ as
 ∗
if k = −Mt , . . . , Nt
 Sk (0) = 0,
ζk (0) =

w∞ (0) = 0, if k = Nt + 1,
(4.46)
ζk (∞) =
83
 ∗
 Sk (∞) = 0,

w∞ (∞) = 1,
if k = −Mt , . . . , Nt
if k = Nt + 1.
Using the above results, the approximate solution wa (z, t) in (4.45) contributes to the
boundary and initial conditions (4.39)-(4.41), as
∂
wa (0, t) =
∂z
∂
wa (1, t) + σ wa (1, t) =
∂z
=
N
z +1
X
N
t +1
X
N
z +1
X
N
t +1
X
j=−Mz −1 k=−Mt
cjk
dξj (0)
ζk (t) = 0 ,
dz
cjk ξj (1)ζk (t) + σ
j=−Mz −1 k=−Mt
N
t +1
X
=
N
t +1
X
0
cjk ξj (1)ζk (t)
j=−Mz −1 k=−Mt
c(Nz +1)k w1 (1)ζk (t) + σ
N
t +1
X
0
c(Nz +1)k w1 (1)ζk (t)
k=−Mt
k=−Mt
N
t +1
X
N
z +1
X
c(Nz +1)k σζk (t) + σ
N
t +1
X
c(Nz +1)k (−1)ζk (t) = 0,
k=−Mt
k=−Mt
and
wa (z, 0) =
N
z +1
X
N
t +1
X
cjk ξj (z)ζk (0) = 0.
j=−Mz −1 k=−Mt
There are several ways to approach the formulation of the fully Sinc-Galerkin method.
For one simple way, we start by rewriting the approximate solution wa (z, t) in (4.45)
as
wa (z, t) = wa1 (z, t) + wa2 (z, t) + wa3 (z, t) + wa4 (z, t) + wa5 (z, t) + wa6 (z, t),
where
wa1 (z, t) =
Nt
X
k=−Mt
c(−Mz −1)k w0 (z)Sk∗ (t),
wa2 (z, t) = c(−Mz −1)(Nt +1) w0 (z)w∞ (t),
84
Nt
X
Nz
X
µ
¶
Sj
cjk
(z)Sk∗ (t),
wa3 (z, t) =
0
φ
j=−Mz k=−Mt
µ ¶
Nz
X
Sj
(z)w∞ (t),
wa4 (z, t) =
cj(Nt +1)
0
φ
j=−M
(4.47)
z
Nt
X
wa5 (z, t) =
c(Mz +1)k w1 (z)Sk∗ (t),
k=−Mt
wa6 (z, t) = c(Mz +1)(Nt +1) w1 (z)w∞ (t).
The unknown coefficients cjk are determined by orthogonalizing the residual with
respect to the set of sinc basis functions (4.42). This leads to the equations
(−Mz − 1 ≤ p ≤ Nz + 1, −Mt ≤ q ≤ Nt + 1)
(Lwa − F, Sp Sq∗ ) = 0,
(4.48)
where the inner product is given by
(f, g) =
Z
∞
0
Z
1
f (z, t)g(z, t)$(z, t)dzdt,
0
and the weight is chosen to be
$(z, t) =
s
Υ0 (t)
.
φ0 (z)
With the expression of the orthogonalized residual in (4.48), this is equivalent to
(Lwa1 , Sp Sq∗ ) + (Lwa2 , Sp Sq∗ ) + (Lwa3 , Sp Sq∗ ) + (Lwa4 , Sp Sq∗ )
+(Lwa5 , Sp Sq∗ ) + (Lwa6 , Sp Sq∗ ) = (F, Sp Sq∗ ).
(4.49)
To each inner product in (4.49) we apply the sinc quadrature rule (3.44) and (3.45).
This leads to the construction of the discrete system that approximates (4.49). From
85
(4.47), we have
Nt
X
Lwa1 (z, t) =
k=−Mt
c(−Mz −1)k
µ
0
w0 (z)(Sk∗ ) (t)
−
00
Av (t)w0 (z)Sk∗ (t)
¶
,
so the inner product involving Lwa1 is
(Lwa1 , Sp Sq∗ )
Nt
X
=
k=−Mt
c(−Mz −1)k
·
¡
0
w0 (Sk∗ ) , Sp Sq∗
¸
¢ ¡
¢
00 ∗
∗
− A v w0 S k , S p S q .
(4.50)
The first inner product in (4.50) is evaluated by first integrating by parts once in the
variable t and then applying the sinc quadrature rule in space and time. First we get
(w0 (Sk∗ ) , Sp Sq∗ ) =
Z
=
Z
0
∞
0
1
0
Z
1
0
w0 (z)(Sk∗ (t)) Sp (z)Sq∗ (t)
·
w0 (z)Sp (z)
p 0
B T1
φ (z)
s
Υ0 (t)
dzdt
(4.51)
φ0 (z)
¸
Z ∞
¡ ∗ √ 0 ¢0
∗
Sk (t) Sq Υ (t)dt dz.
−
0
0
With the same assumptions (3.41) as in Chapter 3, the boundary condition term
B T1 =
(Sk∗ Sq∗
√
¯∞
¯
Υ )(t) ¯¯
0
(4.52)
0
tends to zero, and from (3.40)
¡
Sq∗
√
0
¢0
Υ (t) =
µ
¶
dSq∗ (t) 1 ∗
0
− Sq (t) (Υ (t))3/2 .
dΥ
2
(4.53)
Then applying the sinc quadrature rule, (4.51) becomes
0
(w0 (Sk∗ ) , Sp Sq∗ )
µ
¶
Z ∞
dSq∗ (t) 1 ∗
w0 (z)Sp (z)
0
∗
p 0
=
dz
Sk (t) −
+ Sq (t) (Υ (t))3/2 dt
dΥ
2
φ (z)
0
0
µ
¶
Z ∞
∗
dSq (t) 1 ∗
w0 (zp )
0
∗
Sk (t) −
≈ hz 0
+ Sq (t) (Υ (t))3/2 dt
3/2
(φ (zp ))
dΥ
2
0
¾
½
p
1 (1) 1 (0)
w0 (zp )
− δqk + δqk
≈ h z ht 0
Υ0 (tk ).
(4.54)
3/2
(φ (zp ))
ht
2
Z
1
86
The second inner product in (4.50) is handled by two direct applications of the sinc
quadrature rule (2.11)
00
(Av w0 Sk∗ , Sp Sq∗ )
=
Z
=
Z
∞
0
1
0
1
s
Υ0 (t)
dzdt
φ0 (z)
0
Z ∞
00
p
w0 (z)Sp (z)
p 0
dz
Av (t)Sk∗ (t)Sq∗ (t) Υ0 (t)dt
φ (z)
0
Z
00
w0 (z)Av (t)Sk∗ (t)Sp (z)Sq∗ (t)
00
w (zp ) Av (tk )
≈ hz ht 0 0 3/2 p 0
.
(φ (zp ))
Υ (tk )
So
(Lwa1 , Sp Sq∗ )
(4.55)
½
¾
Nt ·
X
1 (1) 1 (0) p 0
w0 (zp )
c
− δqk + δqk
Υ (tk )
≈ h z ht
0
3/2 (−Mz −1)k
(φ
(z
h
2
p ))
t
k=−Mt
¸
00
Av (tk )
w0 (zp )
.
c(−Mz −1)k p 0
−
(φ0 (zp ))3/2
Υ (tk )
Letting p, q range over all values gives the matrix approximation to (Lwa1 , Sp Sq∗ ),
hz ht {A1 CDt1 + Ds1 CB 1 } ,
(4.56)
with the mz × mz (mz = Mz + Nz + 3) block matrices
A1 ≡
D s1 ≡
£
£
¯
a0−Mz −1 ¯ O mz ×nz
¯
b0−Mz −1 ¯ O mz ×nz
¯
¤
¯ O mz ×1 ,
(4.57)
¯
¤
¯ O mz ×1 ,
the mt × mt (mt = Mt + Nt + 2) block matrices
µ
¶


1
O nt ×1 
 D n t √Υ 0


 Dmt (Av ),
Dt1 ≡ 




O 1×mt


B Tw


,
B1 ≡ 


O 1×mt
(4.58)
87
and the non-square mt × nt (nt = Mt + Nt + 1) matrix
Bw
¾
½
³√ ´
1 (1) 1 (0)
Dnt
Υ0 , −Mt ≤ k ≤ Nt .
≡
− It + It
ht
2
(0)
(1)
Here the mt × nt matrices It , It
(4.59)
are from (2.22)-(2.23), and the mz × 1 column
vectors a0−Mz −1 and b0−Mz −1 have pth component, p = −Mz −1, −Mz , . . . , Nz , Nz +1,
00
[a0−Mz −1 ]p
w (zp )
≡ − 0 0 3/2 ,
(φ (zp ))
The nt × nt diagonal matrix
Dnt
µ
1
√ 0
Υ
¶

[b0−Mz −1 ]p ≡
1
 pΥ0 (t
−Mt )


...
≡ 


and the mt × mt diagonal matrix


Dmt (Av ) ≡ 
w0 (zp )
.
(φ0 (zp ))3/2
Av (t−Mt )
(4.60)

1
p 0
Υ (tNt )



,



...
Av (tNt +1 )

.
Next, the inner product (Lwa5 , Sp Sq∗ ) in (4.49) is similar to (Lwa1 , Sp Sq∗ ), so we get
(Lwa5 , Sp Sq∗ )
¾
½
Nt ·
X
1 (1) 1 (0) p 0
w1 (zp )
c(Mz +1)k − δqk + δqk
Υ (tk )
≈ h z ht
(φ0 (zp ))3/2
ht
2
k=−Mt
¸
00
w1 (zp )
Av (tk )
−
c(Mz +1)k p 0
.
(φ0 (zp ))3/2
Υ (tk )
Letting p, q range over all values gives the matrix approximation to (Lwa5 , Sp Sq∗ ),
hz ht {A5 CDt1 + Ds5 CB 1 } ,
(4.61)
where Dt1 , B1 are in (4.58), the mz × mz block matrices
A5 ≡
D s5 ≡
£
£
¯
O mz ×1 ¯ O mz ×nz
¯
O mz ×1 ¯ O mz ×nz
¯
¤
¯ a1N +1 ,
z
¯ 1
¤
¯ bN +1 ,
z
(4.62)
88
and the mz × 1 column vectors a1Nz +1 and b1Nz +1 have pth component, p = −Mz −
1, −Mz , . . . , Nz , Nz + 1,
00
w1 (zp )
w (zp )
.
(4.63)
≡ − 0 1 3/2 , [b1Nz +1 ]p ≡ 0
(φ (zp ))
(φ (zp ))3/2
µ
¶
0
00
Since Lwa2 (z, t) = c(−Mz −1)(Nt +1) w0 (z)w∞ (t)−Av (t)w0 (z)w∞ (t) , then (Lwa2 , Sp Sq∗ )
[a1Nz +1 ]p
in (4.49) is
(Lwa2 , Sp Sq∗ )
= c(−Mz −1)(Nt +1)
µ
¡
0
w0 w∞ , Sp Sq∗
¢
−
¡
00
Av w0 w∞ , Sp Sq∗
¢
¶
.
Here
¡
0
¢
∗
w0 w∞ , S p S q
=
Z
∞
0
Z
1
0
w0 (z)w∞ (t)Sp (z)Sq∗ (t)
0
s
Υ0 (t)
dzdt,
φ0 (z)
(4.64)
¢
¡
00
Av w0 w∞ , Sp Sq∗ =
Z
∞
0
Z
1
0
00
w0 (z)w∞ (t)Av (t)Sp (z)Sq∗ (t)
s
Υ0 (t)
dzdt.
φ0 (z)
Applying the sinc quadrature rule (2.11), the result is
0
(Lwa2 , Sp Sq∗ )
w0 (zp )
w (tq )
≈ h z ht 0
c(−Mz −1)(Nt +1) p∞ 0
3/2
(φ (zp ))
Υ (tq )
00
w∞ (tq )
w (zp )
.
− hz ht 0 0 3/2 c(−Mz −1)(Nt +1) Av (tq ) p 0
(φ (zp ))
Υ (tq )
Letting p, q range over all values gives the matrix approximation to (Lwa2 , Sp Sq∗ ),
hz ht {A1 CDt2 + Ds1 CB 2 } ,
where A1 , Ds1 are in (4.57), the mt × mt block matrix
(4.65)
89


O nt ×mt

B2 ≡ 


,

aT∞

O nt ×1
O nt ×nt

Dt2 ≡ 

(4.66)


 Dmt (Av ),

bT∞
and the mt × 1 column vectors a∞ and b∞ have q th component, −Mt ≤ q ≤ Nt + 1,
0
[ a∞
w∞ (tq )
[ b ∞ ]q ≡ p 0
.
Υ (tq )
w (tq )
,
]q ≡ p∞ 0
Υ (tq )
(4.67)
From (4.47), wa6 is similar to wa2 , so the inner product is approximated by
Ã
!
0
(t
)
w
q
c(Mz +1)(Nt +1) p∞ 0
(Lwa6 , Sp Sq∗ ) ≈ hz ht
Υ (tq )
Ã
!
¶
µ
00
w∞ (tq )
w1 (zp )
c(Mz +1)(Nt +1) Av (tq ) p 0
− h z ht
.
(φ0 (zp ))3/2
Υ (tq )
µ
w1 (zp )
(φ0 (zp ))3/2
¶
Letting p, q range over all values gives the matrix approximation to (Lwa6 , Sp Sq∗ ),
hz ht {A5 CDt2 + Ds5 CB 2 } ,
(4.68)
where Dt2 and B2 are in (4.66) and A5 and Ds5 are in (4.62).
From(4.47), wa3 is given by
wa3 (z, t) =
Nz
X
Nt
X
j=−Mz k=−Mt
cjk
µ
Sj
φ0
¶
(z)Sk∗ (t) ,
so
Lwa3 (z, t) =
Nz
X
Nt
X
j=−Mz k=−Mt
cjk
µµ
Sj
φ0
¶
0
(z)(Sk∗ ) (t)
µ
Sj
− Av (t) 0
φ
¶00
(z)Sk∗ (t)
¶
.
90
The inner product is
Nz
X
Nt
X
¶
¶
Sj
∗ 0
∗
=
cjk
(Sk ) , Sp Sq
φ0
j=−Mz k=−Mt
¶
µ µ ¶00
Nz
Nt
X
X
Sj
∗
∗
−
cjk Av
Sk , S p Sq .
0
φ
j=−M k=−M
(Lwa3 , Sp Sq∗ )
z
µµ
(4.69)
t
Applying integration by parts in t to the first inner product in (4.69), we get
s
µµ ¶
¶
Z ∞Z 1
Sj
Υ0 (t)
S
(z)
0
0
j
∗
∗
∗
∗
(S
)
,
S
S
=
(S
(t))
S
(z)S
(t)
dzdt
p q
p
0
k
k
q
φ0
φ0 (z)
0
0 φ (z)
·
¸
Z ∞
Z 1
¡ ∗ √ 0 ¢0
Sj (z)Sp (z)
∗
B T2 −
Sk (t) Sq Υ (t)dt dz.
=
(φ0 (z))3/2
0
0
With the same assumption as in (4.51) (because BT2 = BT1 ), the boundary condition
¯∞
√ 0
¯
∗ ∗
term BT2 = (Sk Sq Υ )(t) ¯¯ tends to zero. Then applying the sinc quadrature rule,
0
we get
µµ
Sj
φ0
¶
0
(Sk∗ ) , Sp Sq∗
¶
¶
µ
Z ∞
dSq∗ (t) 1 ∗
Sj (z)Sp (z)
0
∗
=
dz
+
S
(t)
(Υ
(t))3/2 dt
S
(t)
−
0
q
k
3/2
(φ
(z))
dΥ
2
0
¶
µ
Z ∞ 0
dSq∗ (t) 1 ∗
1
0
∗
≈ hz 0
+ Sq (t) (Υ (t))3/2 dt
Sk (t) −
5/2
(φ (zj ))
dΥ
2
0
½
¾
p
1 (1) 1 (0)
1
− δqk + δqk
≈ h z ht 0
Υ0 (tk ).
(4.70)
5/2
(φ (zj ))
ht
2
Z
1
The second inner product in (4.69) is
µ
Av
µ
Sj
φ0
¶00
Sk∗ , Sp Sq∗
¶
=
Z
∞
0
Z
1
0
µ
Sj
Av (t) 0
φ
¶00
(z)Sk∗ (t)Sp (z)Sq∗ (t)
s
Υ0 (t)
dzdt.
φ0 (z)
Applying the sinc quadrature rule, the result is
¶
µ µ ¶00
µ
µ
¶00 ¶
Z ∞
Z 1
√ 0
S
(z)
S
(z)
Sj
p
j
p 0
Sk∗ , Sp Sq∗
=
(Av Sk∗ Sq∗ Υ )(t) BT3 +
dz dt
Av
0
φ0
φ (z)
0
0 φ (z)
(4.71)
91
where
B T3 =
"µ
Sj (z)
φ0 (z)
¶0 Ã
S (z)
pp 0
φ (z)
!
−
µ
Sj (z)
φ0 (z)
¶µ
S (z)
pp 0
φ (z)
¶0 # ¯1
¯
¯ .
¯
(4.72)
0
With the same assumptions (3.41) in Chapter 3, the boundary condition term BT3
tends to zero, and from (2.29)
Ã
!00 µ
¶
Sp (z)
d2 Sp (z) 1
0
p 0
− Sp (z) (φ (z))3/2 .
=
2
dφ
4
φ (z)
So (4.71) becomes
µ
Av
µ
Sj
φ0
¶00
Sk∗ , Sp Sq∗
¶
≈ h z ht
½
Then the inner product (4.69) is
(Lwa3 , Sp Sq∗ )
1 (2) 1 (0)
δ − δpj
h2z pj
4
¾
1
p 0
φ (zj )
Ã
A (t )
pv 0 k
Υ (tk )
!
.
Nt
X
Nz
X
½
¾
cjk
1 (1) 1 (0) p 0
hz ht 0
− δqk + δqk
≈
Υ (tk )
(φ (zj ))5/2
ht
2
j=−Mz k=−Mt
¾
½
Nz
Nt
X
X
cjk
A (t )
1 (2) 1 (0)
pv 0 k .
+
hz ht − 2 δpj + δpj p 0
hz
4
φ (zj ) Υ (tk )
j=−M k=−M
z
t
Letting p, q range over all values gives the matrix approximation to (Lwa3 , Sp Sq∗ ),
hz ht {A3 CDt1 + Ds3 CB 1 } ,
(4.73)
where Dt1 and B1 are defined in (4.58), the mz × mz block matrices
¯
¯
¤
O mz ×1 ¯ Aw ¯ O mz ×1 ,
¶
·
¸
µ
¯
¯ (0)
1
¯ O mz ×1 ,
≡
O mz ×1 ¯ Iz Dnz
(φ0 )5/2
A3 ≡
D s3
£
(4.74)
and the non-square mz × nz matrix
Aw
Ã
!
¾
½
1
1 (2) 1 (0)
Dnz p 0 .
≡
− 2 Iz + Iz
hz
4
φ
(4.75)
92
Lastly, we have wa4 in (4.47) such that
(Lwa4 , Sp Sq∗ )
Nz
X
=
cj(Nt +1)
j=−Mz
Nz
X
−
j=−Mz
µµ
Sj
φ0
¶
0
w∞ , Sp Sq∗
¶
µ
µ ¶00
¶
Sj
∗
cj(Nt +1) Av (t) 0 w∞ , Sp Sq .
φ
(4.76)
Applying the sinc quadrature rule (2.11) to the first inner product in (4.76), the result
is
µµ
Sj
φ0
¶
0
w∞ , Sp Sq∗
¶
=
=
Z
Z
∞
s
1
Υ0 (t)
Sj (z) 0
∗
w∞ (t)Sp (z)Sq (t)
dzdt
0
φ0 (z)
0 φ (z)
s
Z ∞
Υ0 (t)
Sj (z)Sp (z)
0
∗
dz
w∞ (t)Sq (t)
dt
(φ0 (z))3/2
φ0 (z)
0
0
1
0
Z
0
w (t )
1
p∞ 0 q .
≈ h z ht 0
5/2
(φ (zj ))
Υ (tq )
Next, the second inner product in (4.76) is integrated by parts twice in z
µ
Av
µ
Sj
φ0
¶00
w∞ , Sp Sq∗
=
Z
=
Z
∞
0
∞
0
Z
¶
1
0
µ
Sj (z)
Av (t) 0
φ (z)
(Av w∞ Sq∗
√
¶00
Υ0 (t)
dzdt
φ0 (z)
µ
¶00 ¶
Sj (z) Sp (z)
p 0
dz dt,
φ0 (z)
φ (z)
w∞ (t)Sp (z)Sq∗ (t)
µ
Z
Υ )(t) BT4 +
1
0
0
s
where BT4 = BT3 is in (4.72). With the same assumptions (3.41), the boundary
condition term BT4 tends to zero. Applying the sinc quadrature rule (2.11) in space
and time, the result is
µ
Av
µ
Sj
φ0
¶00
w∞ , Sp Sq∗
¶
µ
¶00
Sp (z)
Sj
p 0
=
Υ )(t)dt
dz
0 (z)
φ (z)
0
0 φ
¾
½
1
Av (tq )w∞ (tq )
1 (2) 1 (0)
p 0
δpj − δpj p 0
.
≈ h z ht
2
hz
4
φ (zj )
Υ (tq )
Z
∞
(Av w∞ Sq∗
√
0
Z
1
93
Then the inner product (4.76) is
Nz
X
0
w (tq )
1
cj(Nt +1) p∞ 0
≈ h z ht
0
5/2
(φ (zj ))
Υ (tq )
j=−Mz
¾
½
Nz
X
1
Av (tq )w∞ (tq )
1 (2) 1 (0)
+ h z ht
cj(Nt +1) p 0
− 2 δpj + δpj p 0
hz
4
φ (zj )
Υ (tq )
j=−Mz
(Lwa4 , Sp Sq∗ )
Letting p, q range over all values gives the matrix approximation to (Lwa4 , Sp Sq∗ ),
hz ht {A3 CDt2 + Ds3 CB 2 } ,
(4.77)
where A3 and Ds3 are in (4.74) and Dt2 and B2 are in (4.66).
Applying the sinc quadrature rule (2.10), the inner product in the right-hand-side
of (4.49) leads to
∞
1
s
Υ0 (t)
dzdt
φ0 (z)
0
0
1
1
p
F
(z
,
t
)
.
≈ h z ht 0
p
q
(φ (zp ))3/2
Υ0 (tq )
(F, Sp Sq∗ ) =
Z
Z
F (z, t)Sp (z)Sq∗ (t)
(4.78)
Letting p, q range over all values gives the matrix approximation to (F, Sp Sq∗ ),
h z h t D mz
µ
1
0 3/2
(φ )
¶
F D nt
µ
1
√ 0
Υ
¶
,
(4.79)
where the pq th -entry of F ( −Mz − 1 ≤ p ≤ Nz + 1 and −Mt ≤ q ≤ Nt + 1) , contains
the point evaluation of the function F (z, t) or F (zp , tq ).
Finally, we can determine the discrete system by substituting the expressions
(4.56), (4.61), (4.65), (4.68), (4.73), (4.77), and (4.79) into (4.49) and multiplying
both sides by (hz ht )−1 . With simplification, the discrete system that represents the
94
initial-boundary-value problem (4.38)-(4.41) becomes
A1 CDt1 + Ds1 CB 1 + A1 CDt2 + Ds1 CB 2 + A3 CDt1 + Ds3 CB 1 + A3 CDt2
+Ds3 CB 2 + A5 CDt1 + Ds5 CB 1 + A5 CDt2 + Ds5 CB 2
µ
¶
µ
¶
1
1
F D nt √ 0
= D mz
(φ0 )3/2
Υ
which gives
A1 C(Dt1 + Dt2 ) + Ds1 C(B1 + B2 ) + A3 C(Dt1 + Dt2 ) + Ds3 C(B1 + B2 )
¶
µ
¶
µ
1
1
F D nt √ 0 .
+A5 C(Dt1 + Dt2 ) + Ds5 C(B1 + B2 ) = Dmz
(φ0 )3/2
Υ
So, the result is
(A1 + A3 + A5 )C(Dt1 + Dt2 ) + (Ds1 + Ds3 + Ds5 )C(B1 + B2 )
µ
µ
¶
¶
1
1
= D mz
F D nt √ 0
(φ0 )3/2
Υ
which leads to
ACDt + Ds CB = F,
(4.80)
where the mz × mz block matrices are
A ≡ A 1 + A3 + A5
≡
+
£
£
¯
a0−Mz −1 ¯ O mz ×nz
¯
¤
¯ O mz ×1
¯
¯
¤
O mz ×1 ¯ Aw ¯ O mz ×1
95
+
≡
£
£
¯
O mz ×1 ¯ O mz ×nz
¯
¤
¯ a1N +1
z
¯
¯
¤
a0−Mz −1 ¯ Aw ¯ a1Nz +1 ,
(4.81)
D s ≡ D s1 + D s3 + D s5
¯
b0−Mz −1 ¯ O mz ×nz
µ
·
¯ (0)
¯
Iz D n z
+
O mz ×1
≡
£
+
£
≡
·
¯
O mz ×1 ¯
¯
b0−Mz −1 ¯
¯
¤
¯ O mz ×1
¶
¸
¯
1
¯
O mz ×1
(φ0 )5/2
¯
¤
O mz ×nz ¯ b1Nz +1
¶
¸
µ
¯ 1
1
(0)
¯ bN +1 ,
Iz D n z
z
(φ0 )5/2
and
F≡
D mz
µ
1
0 3/2
(φ )
¶
F D nt
µ
1
√ 0
Υ
¶
.
(4.82)
Also the mt × mt block matrices
Dt ≡ D t1 + D t2
µ
¶

1
 D n t √Υ 0

≡ 



O nt ×nt

+ 


 Dnt

≡ 


µ
bT∞
¶
1
√ 0
Υ

O nt ×1 

 Dmt (Av )


O 1×mt

O nt ×1

 Dmt (Av )

O nt ×1
bT∞



 Dmt (Av ),


(4.83)
96
B ≡ B1 + B2

B Tw

≡ 

O 1×mt


 
+
 
O nt ×mt
aT∞


 
≡
 
B Tw
aT∞


,

where B w , a0−Mz −1 , b0−Mz −1 , a1Nz +1 , b1Nz +1 , a∞ , b∞ , and Aw are defined in (4.59),
(4.60), (4.63), (4.67), and (4.75), respectively.
There are various methods for solving the generalized Sylvester equation (4.80).
They are described in [12]. Using Theorem 6 in Chapter 2, (4.80) is algebraically
equivalent to the system
Gco(C) = co(F),
(4.84)
where the matrix G involving Kronecker products is given by an (mz mt ) × (mz mt )
matrix
G = DtT ⊗ A + B T ⊗ Ds ,
and co(C), co(F) are (mz mt ) × 1 column vectors.
Parameter Selections for the Fully Sinc-Galerkin Method
The matrices that comprise the discrete system in the Sinc-Galerkin method are
full matrices. More sinc grid points lead to larger matrices and make for an expensive
computation. Some cases found in [12] show how to choose an appropriate sinc grid
in space and time. If u(z, t) is a real solution that satisfies the condition
¯
¯
¯u(z, t)¯ ≤ Cz αs +1/2 (1 − z)βs +1/2 tγ+1/2 t−δ+1/2
(4.85)
97
for (z, t) ∈ (0, 1) × (0, ∞), we should make the selections
and
¯¸
·¯
¯
¯ αs
Nz = ¯¯ Mz + 1¯¯
βs
,
hs =
¯¸
·¯
¯ αs
¯
Mt = ¯¯ Mz + 1¯¯
γ
µ
,
If the exact solution satisfies the condition
πd
α s Mz
¶1/2
hs ≡ hz = ht ,
,
¯¸
·¯
¯
¯ αs
Nt = ¯¯ Mz + 1¯¯ .
δ
¯
¯
¯u(z, t)¯ ≤ Cz αs +1/2 (1 − z)βs +1/2 tγ+1/2 e−δt
(4.86)
(4.87)
for (z, t) ∈ (0, 1) × (0, ∞), we should make the selections
and
¯¸
·¯
¯ αs
¯
¯
Nz = ¯ Mz + 1¯¯
βs
¯¸
·¯
¯ αs
¯
¯
Mt = ¯ Mz + 1¯¯
γ
,
hs =
,
µ
πd
α s Mz
¶1/2
, hs ≡ hz = ht ,
¯¸
·¯
´
¯ 1 ³ αs
¯
¯
Nt = ¯ ln
Mz hs + 1¯¯ .
hs
δ
(4.88)
If the exact solution satisfies the condition
¯
¯
¯u(z, t)¯ ≤ Cz α (1 − z)β tγ e−δt
for (z, t) ∈ (0, 1) × (0, ∞), we should make the selections
¯¸
¸
·
¸
·¯
´
¯
¯
¯
¯α
¯ 1 ³α
¯α
Mz h + 1¯¯ ,
Nz = ¯ Mz + 1¯ , Mt = ¯ Mz + 1¯ , Nt = ¯¯ ln
β
γ
h
δ
·
where h ≡ hz = ht , and
h=
µ
πd
αMz
¶1/2
.
(4.89)
98
In general, the exact solution is unknown, so one must determine these parameters
via numerical experimentation for a given problem. Thus we assume α = β = γ = 1
√
in (4.89), choose d = π/2 which leads to h = π/ 2Mz . Thus Mz = Nz = Mt . Then
we experiment and determine Nt , which gives the sinc grid in time for the infinite
interval.
Numerical Examples for Time-Independent Boundary Conditions
The examples in this section illustrate various features of different sinc grids, timedependent eddy viscosities Av (t), and parameter selections σ in (4.38)-(4.41). Each
successive example is devised to test every aspect of the method on problems with
known solutions. This will lend numerical evidence to the reliability of this method
when applied to the realistic oceanographic problems in Chapter 5. To illustrate the
performance of the method, we redefine kUS k, kVS k and kES k in (3.63) for reporting
errors and convergence results on the sinc grid S with h ≡ hz = ht as
S = {(zj , tk ) : zj =
ejh
, tk = ekh , −Mz − 1 ≤ j ≤ Nz + 1, −Mt ≤ k ≤ Nt + 1},
1 + ejh
(4.90)
kUS k = max {U0 |Ua (zj , tk ) − U (zj , tk )|} ,
S
kVS k = max {U0 |Va (zj , tk ) − V (zj , tk )|} ,
S
and
kES k = max {kUS k, kVS k} .
(4.91)
99
We also report errors and convergence results on the uniform grid U with step size
lz = .01 and lt = 0.1 as
U = {(zm , tn ) : zm = mlz , tn = nlt , m = 0, 1, . . . , 100, n = 0, 1, . . . , 100}, (4.92)
kUU k = max {U0 |Ua (zm , tn ) − U (zm , tn )|} ,
U
kVU k = max {U0 |Va (zm , tn ) − V (zm , tn )|} ,
U
and
kEU k = max {kUU k, kVU k} .
(4.93)
Here Ua (z, t) and Va (z, t) are computed from approximate solutions and U (z, t) and
V (z, t) are computed from the exact solutions. If the solution w(z, t) = u(z, t)+iv(z, t)
is real (v(z, t) ≡ 0) then kVS k and kVU k are also computed by comparing a SinGalerkin approximate solution Va (z, t) to the V (z, t) ≡ 0 zero solution. For visual
clarity, all three-dimensional graphs of approximate solutions are plotted by using the
data from a uniform grid with step size lz = 0.025 and lt = 0.1 as
U = {(zm , tn ) : zm = mlz , tn = nlt , m = 0, 1, . . . , 40, n = 0, 1, . . . , 100}. (4.94)
Two-dimensional graphs of approximate solutions are plotted by using the sinc grid
S t = {tk : tk = ekh , −Mt ≤ k ≤ Nt + 1}
(4.95)
and the uniform grid with step size lt = 0.1 as
U t = {tn : tn = nlt , n = 0, 1, . . . , 100}.
(4.96)
100
Example 9. First consider a constant function, Av (t) ≡ 1, with the parameter σ = 0.
The initial-boundary-value problem (4.38)-(4.41) is a heat equation with Neumann
boundary condition at z = 0 and Dirichlet boundary condition at z = 1,
∂w(z, t) ∂ 2 w(z, t)
= F (z, t), 0 < z < 1, 0 < t,
−
∂t
∂z 2
∂w(0, t)
= 0, 0 < t
∂z
w(1, t) = 0, 0 < t
w(z, 0) = 0, 0 < z < 1.
This problem has been addressed in [1] and [11]. With the forcing function F (z, t) =
(1 − z)ez−t (1 − t) + (1 + z)tez−t , the exact real solution is w(z, t) = (1 − z)tez−t . The
discrete system is given by (4.80) and we use the parameter choices α = β = γ = 1 in
(4.90) except for the choice of Nt , which is determined by numerical experiment. The
results are reported in Table 8, Table 9, and graphs are shown in Figure 30, Figure
31, Figure 32, and Figure 33, respectively.
Table 8 shows the solution errors on
the sinc grid S with Mz = Nz = Mt and 1 ≤ Nt ≤ Mz . The sinc grid was doubled
(by doubling Mz ) until Mz = 16. Also the errors on the uniform grid U which are
generated by using (4.45) are reported. We will not report kUU k and kVU k which are
similar to the reported kUS k and kVS k. There is little difference after
Mz
≤ Nt . So a
2
reasonable choice for choosing Nt is Nt = (1/2)Mz . Remember, Nt is the number of
sinc grid points in the right-hand side of the sinc time domain (1 < t). Having settled
on this, Table 9 shows the errors with Mz = Nz = Mt = 2Nt for Mz = 4, 8, 16, 32.
101
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 4.544e-03
1.111 4.529e-03
1.111 4.542e-03
1.111 4.536e-03
0.785 1.093e-03
0.785 1.087e-03
0.785 1.089e-03
0.785 1.090e-03
0.785 1.090e-03
0.785 1.090e-03
0.785 1.090e-03
0.785 1.090e-03
0.555 1.040e-02
0.555 1.485e-03
0.555 1.105e-04
0.555 1.084e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
0.555 1.083e-04
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
4.544e-03
4.529e-03
4.542e-03
4.536e-03
1.093e-03
1.087e-03
1.089e-03
1.090e-03
1.090e-03
1.090e-03
1.090e-03
1.090e-03
1.040e-02
1.485e-03
1.105e-04
1.084e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.083e-04
1.608e-02
1.579e-02
1.567e-02
1.562e-02
3.885e-02
2.931e-03
2.969e-03
2.979e-03
2.980e-03
2.979e-03
2.978e-03
2.978e-03
1.649e-01
2.889e-02
4.216e-04
2.892e-04
2.884e-04
2.871e-04
2.862e-04
2.856e-04
2.852e-04
2.849e-04
2.848e-04
2.847e-04
2.846e-04
2.846e-04
2.845e-04
2.845e-04
Table 8. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choice of Nt for Example 9 with Av (t) ≡ 1 and the parameter σ = 0 .
102
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
h
kUS k
kVS k
1.111 4.529e-03 0.000e+00
0.785 1.090e-03 0.000e+00
0.555 1.083e-04 0.000e+00
0.393 2.693e-06 0.000e+00
kES k
kEU k
4.529e-03 1.579e-02
1.090e-03 2.979e-03
1.083e-04 2.856e-04
2.693e-06 4.085e-06
Table 9. Errors on the sinc grid S and the uniform grid U for the choice Mz = Nz =
Mt = 2Nt for Example 9 with Av (t) ≡ 1 and the parameter σ = 0.
The approximate solution is graphed in Figure 30 using the values generated on the
sinc grid S.
The approximate solution is graphed in Figure 31 using the values on
the uniform grid U in (4.94).
The approximate solution is graphed in Figure 32
using the values generated on the sinc grid S t . This time plot is graphed at the sea
surface, z = 0, middle ocean depth, z = .50, and the seabed, z = 1. The plot is also
graphed at z = .15895 which is the sinc gridpoint nearest the Ekman depth DE = .2.
For ease of reference we shall refer to the sinc gridpoint as z = .16. The values of
z = 0, .16, .50 and 1 correspond to 0 (sea surface), DE (Ekman depth), (1/2)D0
(middle ocean depth), and D0 (seabed), respectively. This is done in all the following
examples. The approximate solution is graphed in Figure 33 using the values on the
uniform grid U t . This time plot is graphed for z = 0, .20, .50 and 1. The values
of z = 0, .20, .50 and 1 correspond to 0 (sea surface), DE (Ekman depth), (1/2)D0
(middle ocean depth), and D0 (seabed), respectively.
103
0.2
0.4
0.4
0.35
0.3
wa (z, t)
0.25
0.2
0.15
0.1
0.05
0
1
0.8
10
0.6
PSfrag replacements
8
6
0.4
z
2
0.2
0
2
0
t
0
Figure 30. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 9 where Av (t) ≡ 1 and σ = 0 with Mz = Nz = Mt = 16 , Nt = 8.
0.2
0.4
0.4
0.35
0.3
wa (z, t)
0.25
0.2
0.15
0.1
0.05
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 31. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 9 where Av (t) ≡ 1 and σ = 0 with Mz = Nz = Mt = 16, Nt = 8.
0.2
0.4
104
0.6
0.8
0.4
0.01
z=0
z = .16
z = .50
z=1
0.35
0.3
1
wa (z, t)
0.25
PSfrag replacements
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 32. The
0.2 approximate solution wa (z, t) when z = 0, .16, .50, and 1 on the sinc
grid S t for Example
9 with Mz = Nz = Mt = 16, and Nt = 8.
0.4
0.6
0.8
0.4
0.01
z=0
z = .20
z = .50
z=1
0.35
0.3
wa (z, t)
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 33. The approximate solution of wa (z, t) at z = 0, .20, .50, and 1 on the
uniform grid U t for Example 9 with Mz = Nz = Mt = 16, and Nt = 8.
105
Example 10. Now choose Av (t) ≡ 1 and the parameter σ = 1. The initial-boundaryvalue problem (4.38)-(4.41) is a heat equation with Neumann boundary condition at
z = 0 and mixed boundary condition at z = 1 given as
∂w(z, t) ∂ 2 w(z, t)
−
= F (z, t), 0 < z < 1, 0 < t
∂t
∂z 2
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
= 0, 0 < t
w(1, t) +
∂z
w(z, 0) = 0, 0 < z < 1.
This type of problem is treated in [1]. With the forcing function F (z, t) = (z − 2 +
e−z )e−t −e−z (1−e−t ), the exact solution is w(z, t) = (z−2+e−z )(1−e−t ). The discrete
system is given by (4.80). We use the parameter choices α = β = γ = 1 in (4.90) and
experimentally determine Nt . The results are reported in Table 10, Table 11, and
Figure 34, Figure 35, Figure 36, and Figure 37, respectively. Table 10 again suggests
that we choose Nt = (1/2)Mz . Table 11 shows errors with Mz = Nz = Mt = 2Nt
for Mz = 4, 8, 16, 32. The results are slightly better than those in Example 9
which had a Dirichlet boundary condition at z = 1. The approximate solution is
shown in Figure 34 using values generated on the sinc grid S. The graph in Figure
35 is the approximate solution wa (z, t) on the uniform grid U . The time plot of the
approximate solution wa (z, t) in Figure 36 is graphed for the values z = 0, .16, .50,
and 1 on the sinc grid S t . Figure 37 shows the time plot of the approximate solution
wa (z, t) for z = 0, .20, .50, and 1 on the uniform grid U t .
106
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 2.016e-03
1.111 3.271e-03
1.111 3.117e-03
1.111 3.139e-03
0.785 6.776e-03
0.785 2.708e-03
0.785 4.639e-04
0.785 5.176e-04
0.785 4.518e-04
0.785 4.661e-04
0.785 4.631e-04
0.785 4.639e-04
0.555 7.117e-03
0.555 6.068e-03
0.555 3.015e-03
0.555 1.219e-03
0.555 4.918e-04
0.555 1.710e-04
0.555 7.590e-05
0.555 3.220e-05
0.555 3.848e-05
0.555 3.632e-05
0.555 3.710e-05
0.555 3.682e-05
0.555 3.692e-05
0.555 3.688e-05
0.555 3.690e-05
0.555 3.689e-05
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
2.016e-03
3.271e-03
3.117e-03
3.139e-03
6.776e-03
2.708e-03
4.639e-04
5.176e-04
4.518e-04
4.661e-04
4.631e-04
4.639e-04
7.117e-03
6.068e-03
3.015e-03
1.219e-03
4.918e-04
1.710e-04
7.590e-05
3.220e-05
3.848e-05
3.632e-05
3.710e-05
3.682e-05
3.692e-05
3.688e-05
3.690e-05
3.689e-05
6.034e-03
4.701e-03
3.771e-03
3.968e-03
7.985e-02
6.174e-03
1.653e-03
5.744e-04
6.904e-04
5.430e-04
5.939e-04
5.716e-04
1.407e-01
7.149e-02
5.696e-03
1.704e-03
6.485e-04
2.539e-04
1.289e-04
7.109e-05
4.715e-05
4.139e-05
3.763e-05
3.709e-05
3.693e-05
3.693e-05
3.689e-05
3.690e-05
Table 10. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt in Example 10 with Av (t) ≡ 1 and the parameter σ = 1.
107
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
kVS k
kES k
1.111 3.271e-03 0.000e+00
0.785 5.176e-04 0.000e+00
0.555 3.220e-05 0.000e+00
0.393 1.217e-06 0.000e+00
kEU k
3.271e-03 4.701e-03
5.176e-04 5.744e-04
3.220e-05 7.109e-05
1.217e-06 2.701e-06
Table 11. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt of Example 10 with Av (t) ≡ 1 and the parameter σ = 1.
PSfrag replacements
0
0
wa (z, t)
-0.2
-0.4
-0.6
-0.8
-1
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 34. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 10 where Av (t) ≡ 1 and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
108
0
0
-0.2
1
wa (z, t)
PSfrag replacements
-0.4
-0.6
-0.8
-1
1
0.8
10
0.6
8
6
0.4
z
4
0.2
t
2
0
0
0.2
0.4
0.6 The graph of the approximate solution w a (z, t) on the uniform grid U for
Figure 35.
Example0.810 where Av (t) ≡ 1 and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
0
z=0
z = .16
z = .50
z=1
-0.1
-0.2
-0.3
wa (z, t)
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
−1
0
1
2
3
4
5
6
7
8
9
10
t
Figure 36. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid
S t for Example 10 with Mz = Nz = Mt = 16, Nt = 8 .
0.6
0.8
109
0
z=0
z = .20
z = .50
z=1
-0.1
-0.2
-0.3
wa (z, t)
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
−1
0
1
2
3
4
5
6
7
8
9
10
t
Figure 37. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 10 with Mz = Nz = Mt = 16, Nt = 8.
Example 11. In this example we choose Av (t) =
t+1
and the parameter σ = 1. This
t+2
is a heat equation with a time-dependent coefficient, Neumann boundary condition at
z = 0 and mixed boundary condition at z = 1. So the initial-boundary-value problem
(4.38)-(4.41) becomes
∂
∂w(z, t)
−
∂t
∂z
µµ
¶
¶
t + 1 ∂w(z, t)
= F (z, t), 0 < z < 1, 0 < t
t+2
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
w(1, t) +
= 0, 0 < t
∂z
w(z, 0) = 0, 0 < z < 1.
With the forcing function given by
F (z, t) =
(z 2 − 2z 3 + z 4 )
2t
− (1 − 6z + 6z 2 )
,
2
(1 + t)
(1 + t)
110
the exact solution is
w(z, t) = z 2 (1 − z)2
t
.
(1 + t)
The discrete system is given by (4.80) and we use the parameter choices α = β = γ = 1
in (4.90) and experimentally determine Nt . The results are reported in Table 13, Table
12, and Figure 38, Figure 39, Figure 40, and Figure 41, respectively. Table 13 again
suggests that we choose Nt = (1/2)Mz . Table 12 shows errors on the sinc grid S
and the uniform grid U for Mz = 4, 8, 16, 32. The accuracy on the uniform grid is
slightly better than the accuracy on the sinc grid. The approximate solution is shown
in Figure 38 using values generated on the sinc grid S. The approximate solution
wa (z, t) is graphed in Figure 39 on the uniform grid U . The approximate solution
wa (z, t) is graphed in Figure 40 using the values generated on the sinc grid S t . This
time plot is graphed for z = 0, .16, .50 and 1.
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
h
kUS k
kVS k
1.111 2.021e-02 0.000e+00
0.785 4.124e-03 0.000e+00
0.555 3.949e-04 0.000e+00
0.393 1.227e-05 0.000e+00
kES k
kEU k
2.021e-02 1.799e-02
4.124e-03 3.560e-03
3.949e-04 3.334e-04
1.227e-05 1.026e-05
Table 12. Errors on the sinc grid S and the uniform grid U for the choices Mz =
t+1
Nz = Mt = 2Nt for Example 11 with Av (t) =
and σ = 1.
t+2
111
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 1.767e-02
1.111 2.021e-02
1.111 2.109e-02
1.111 2.135e-02
0.785 2.784e-03
0.785 3.605e-03
0.785 3.960e-03
0.785 4.124e-03
0.785 4.198e-03
0.785 4.233e-03
0.785 4.249e-03
0.785 4.256e-03
0.555 1.935e-04
0.555 2.735e-04
0.555 3.286e-04
0.555 3.589e-04
0.555 3.760e-04
0.555 3.859e-04
0.555 3.916e-04
0.555 3.949e-04
0.555 3.968e-04
0.555 3.979e-04
0.555 3.985e-04
0.555 3.989e-04
0.555 3.991e-04
0.555 3.992e-04
0.555 3.993e-04
0.555 3.993e-04
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
1.767e-02
2.021e-02
2.109e-02
2.135e-02
2.784e-03
3.605e-03
3.960e-03
4.124e-03
4.198e-03
4.233e-03
4.249e-03
4.256e-03
1.935e-04
2.735e-04
3.286e-04
3.589e-04
3.760e-04
3.859e-04
3.916e-04
3.949e-04
3.968e-04
3.979e-04
3.985e-04
3.989e-04
3.991e-04
3.992e-04
3.993e-04
3.993e-04
1.797e-02
1.799e-02
1.799e-02
1.799e-02
3.073e-03
3.573e-03
3.558e-03
3.560e-03
3.560e-03
3.560e-03
3.560e-03
3.560e-03
2.353e-04
2.939e-04
3.329e-04
3.336e-04
3.333e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
3.334e-04
Table 13. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
t+1
various choices of Nt for Example 11 with Av (t) =
and σ = 1.
t+2
0
112
0.1
0.08
wa (z, t)
0.06
0.04
0.02
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
PSfrag replacements
Figure 38. 0 The graph of the approximate solution wa (z, t) on the sinc grid S for
t+1
Example 11 where Av (t) =
and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
t+2
0.1
0.08
wa (z, t)
0.06
0.04
0.02
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 39. The graph of the approximate solution wa (z, t) on the uniform grid U for
t+1
and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
Example 11 where Av (t) =
t+2
113
0.06
0.05
wa (z, t)
0.04
0.07
z
z
z
z
=0
= .16
= .50
=1
0.03
0.02
0.01
0
PSfrag replacements
0
1
2
3
4
5
6
7
8
9
10
t
Figure 40. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
t+1
for Example 11 where Av (t) =
and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
t+2
0.06
0.05
wa (z, t)
0.04
0.07
0.03
0.02
z
z
z
z
0.01
0
0
1
2
3
4
5
6
7
=0
= .20
= .50
=1
8
9
10
t
Figure 41. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform grid
t+1
U t for Example 11 where Av (t) =
and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
t+2
114
Solving the Problem with Time-Dependent Boundary Conditions
The previous section showed how to develop the linear discrete system for solving
our partial differential equation with time-independent boundary conditions. More
generally, a Sinc-Galerkin method can be developed to produce a discrete system for
an initial-boundary-value problem with time-dependent boundary conditions given in
(4.34)-(4.37). So consider
Hw(z, t) − ifc k2 w(z, t) = F (z, t), 0 < z < 1, 0 < t
(4.97)
where
µ
¶
fc ∂
∂w(z, t)
∂w(z, t)
− 2
Hw(z, t) ≡
Av (t)
.
∂t
2κ ∂z
∂z
The time-dependent boundary conditions are
∂w(0, t)
= 0, 0 < t,
∂z
w(1, t) + σAv (t)
∂w(1, t)
= 0, 0 < t,
∂z
(4.98)
(4.99)
with the initial condition
w(z, 0) = 0, 0 < z < 1.
(4.100)
Note that for (4.97)-(4.100) an approach similar to that in (4.38)-(4.41) may be used.
However, we are not able to choose single variable functions to be boundary basis functions that will be appropriate, at the endpoints, for the boundary conditions
115
(4.98)-(4.99). In a simple way, we give two-variable functions for an additional boundary basis function as (compare to (4.43))
w0 (z) = (2z + 1)(1 − z)2 ,
w1 (z, t) = (1 − z)z 2 + σAv (t)(3 − 2z)z 2 ,
(4.101)
and an additional basis function for the temporal space (same as (4.44))
w∞ (t) =
t
.
1+t
(4.102)
A fully Sinc-Galerkin method to approximate the solution to (4.97)-(4.100) with the
mixed time-dependent boundary condition in (4.99) can be formulated as follows.
©
ª
With the product basis functions Sp (z)Sq∗ (t) in (4.42) where −Mz −1 ≤ p ≤ Nz +1,
and −Mt ≤ q ≤ Nt + 1, define the approximate solution
wa (z, t) =
N
z +1
X
N
t +1
X
cjk ξj (z, t)ζk (t)
(4.103)
j=−Mz −1 k=−Mt
where

w0 (z),






 µS ¶
j
ξj (z, t) =
(z),

φ0






w1 (z, t),
ζk (t) =
 ∗
 Sk (t),

w∞ (t),
if j = −Mz − 1
if j = −Mz , . . . , Nz
(4.104)
if j = Nz + 1,
if k = −Mt , . . . , Nt
if k = Nt + 1.
The unknown coefficients cjk will be stored in the mz ×mt unknown coefficient matrix
C = [cjk ].
116
Initially, we have to show that the approximate solution (4.103) satisfies the
boundary conditions (4.98)-(4.99) and the initial condition (4.100). We take the first
partial derivative with respect to z as
N
N
t +1
z +1
X
X
∂ξj (z, t)
∂wa (z, t)
cjk
=
ζk (t),
∂z
∂z
j=−M −1 k=−M
z
(4.105)
t
where
 0
w0 (z),







 µ ¶0
∂ξj (z, t)  Sj (z),
=
φ0

∂z







 ∂w1 (z, t) ,
∂z
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1.
The expressions in (4.106) are evaluated at z =
 0
w0 (0) = 0,







µ ¶0


Sj
∂ξj (0, t)
(0) = 0,
=
φ0

∂z







 ∂w1 (0, t) = 0,
∂z
and
 0

 w0 (1) = 0,





 µ ¶0
∂ξj (1, t)  Sj (1) = 0,
=
φ0

∂z







 ∂w1 (1, t) = −1,
∂z
Also

w0 (1) = 0,






 µS ¶
j
ξj (1, t) =
(1) = 0,
0

φ






w1 (1, t) = σAv (t),
0 and 1. The results are
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1 ,
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1.
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1.
(4.106)
117
Lastly, as before,
ζk (0) =
 ∗
 Sk (0) = 0,

ζk (∞) =
if k = −Mt , . . . , Nt
w∞ (0) = 0,
if k = Nt + 1,
 ∗
 Sk (∞) = 0,

if k = −Mt , . . . , Nt
w∞ (∞) = 1,
if k = Nt + 1.
From the above results, the time-dependent boundary conditions (4.98)-(4.99) and
initial condition (4.100) are
∂
wa (0, t) =
∂z
N
z +1
X
N
t +1
X
cjk
j=−Mz −1 k=−Mt
∂
wa (1, t) + σAv (t) wa (1, t) =
∂z
∂ξj (0, t)
ζk (t) = 0,
∂z
N
t +1
X
N
z +1
X
j=−Mz −1 k=−Mt
N
z +1
X
+ σAv (t)
cjk ξj (1, t)ζk (t)
N
t +1
X
cjk
j=−Mz −1 k=−Mt
= σAv (t)
N
t +1
X
∂ξj (1, t)
ζk (t)
∂z
c(Nz +1)k ζk (t)
k=−Mt
+ σAv (t)
N
t +1
X
(−1)c(Nz +1)k ζk (t)
k=−Mt
= (σAv (t) − σAv (t))
N
t +1
X
c(Nz +1)k ζk (t) = 0.
k=−Mt
and
wa (z, 0) =
N
z +1
X
N
t +1
X
cjk ξj (z)ζk (0) = 0.
j=−Mz −1 k=−Mt
In order to use the same approach as in the previous section, we now rewrite the
additional boundary basis function w1 (z, t) in (4.43) as
118
w1 (z, t) = wg (z) + σAv (t)wh (z),
(4.107)
where
wg (z) = (1 − z)z 2 ,
wh (z) = (3 − 2z)z 2 .
The approximate solution wa (z, t) in (4.103) can be written in the separated form
wa (z, t) = wc (z, t) + wb (z, t),
(4.108)
where
wc (z, t) =
N
z +1
X
N
t +1
X
cjk ξj∗ (z)ζk (t),
j=−Mz −1 k=−Mt
wb (z, t) = σAv (t)
N
t +1
X
c(Nz +1)k wh (z)ζk (t),
(4.109)
k=−Mt
and the basis functions ξj∗ (z) are

w0 (z),






 µS ¶
j
∗
(z),
ξj (z) =
0

φ






wg (z),
if j = −Mz − 1
if j = −Mz , . . . , Nz
if j = Nz + 1 ,
with ζk (t) in (4.104).
The unknown coefficients cjk are determined by orthogonalizing the residual with
respect to the set of sinc basis functions in (4.42). This yields the discrete system
(Hwa − ifc k2 wa − F, Sp Sq∗ ) = 0,
(4.110)
119
or
(Hwc , Sp Sq∗ ) + (Hwb , Sp Sq∗ ) − (ifc k2 wa , Sp Sq∗ ) = (F, Sp Sq∗ ),
(4.111)
where −Mz − 1 ≤ p ≤ Nz + 1 and −Mt ≤ q ≤ Nt + 1 . The inner product is given by
(f, g) =
Z
∞
0
Z
1
f (z, t)g(z, t)
0
s
Υ0 (t)
dzdt.
φ0 (z)
The first inner product (Lwc , Sp Sq∗ ) in (4.111) is similar to the inner product in
(4.48). Letting p, q range over all values gives the matrix approximation to (Hw c , Sp Sq∗ ),
hz ht {Ag CDt + Dg CB} ,
(4.112)
where the mz × mz block matrices are
¯
¯
¤
fc £ 0
¯ ag
¯
A
,
a
w
−Mz −1
Nz +1
2
2κ
·
¶
¸
µ
¯ g
¯ (0)
1
0
¯
¯
≡
b−Mz −1
Iz D n z
bNz +1 ,
(φ0 )5/2
Ag ≡
Dg
(4.113)
the mt × mt block matrices Dt and B are in (4.83) and the mz × 1 column vectors
a0−Mz −1 and b0−Mz −1 are in (4.60). The mz × 1 column vectors agNz +1 and bgNz +1 have
pth component, p = −Mz − 1, −Mz , . . . , Nz , Nz + 1,
00
[agNz +1 ]p
wg (zp )
≡− 0
,
(φ (zp ))3/2
[bgNz +1 ]p ≡
wg (zp )
.
(φ0 (zp ))3/2
(4.114)
Next, from (4.109), we have
Hwb (z, t) = σ
N
t +1
X
k=−Mt
¶
µ
fc 2
00
0
c(Nz +1)k wh (z)(Av (t)ζk (t)) − 2 Av (t)ζk (t)wh (z) ,
2κ
120
so the second inner product in (4.111) is
(Lwb , Sp Sq∗ )
= σ
Nt
X
c(Nz +1)k
k=−Mt
½
¡
wh (Av ζk )
0
, Sp Sq∗
¢
¢
fc ¡
00
− 2 A2v ζk wh , Sp Sq∗
2κ
¾
(4.115)
¾
½
¢
¡
¢
fc ¡ 2
00
0
∗
∗
+ σc(Nz +1)(Nt +1) wh (Av w∞ ) , Sp Sq − 2 Av w∞ wh , Sp Sq .
2κ
Applying integration by parts once in t to the first inner product in (4.115), the result
is
(wh Av ζk , Sp Sq∗ ) =
Z
=
Z
¡
¢0
∞
0
1
0
Z
1
0
µ
wh (z)Sp (z)
p 0
B T5
φ (z)
s
Υ0 (t)
dzdt
φ0 (z)
¶
Z ∞
¡ ∗ √ 0 ¢0
−
Av (t)ζk (t) Sq Υ (t)dt dz.
¢0
wh (z) Av (t)ζk (t) Sp (z)Sq∗ (t)
¡
0
Using the same assumption as in (4.51), the boundary condition term
B T5 =
Av (t)ζk (t)Sq∗ (t)
p
¯∞
¯
Υ (t) ¯¯
0
(4.116)
0
tends to zero. Applying the sinc quadrature rule in space and time, then leads to
Z
¡ √ ¢0
wh (z)Sp (z) ∞
p 0
−Av (t)ζk (t) Sq∗ Υ0 (t)dtdz (4.117)
φ (z)
0
0
½
¾
p 0
1 (1) 1 (0)
wh (zp )
−
≈ h z ht 0
δ
δ
Υ (tk ).
+
A
(t
)
v
k
qk
qk
(φ (zp ))3/2
ht
2
¡
¢0
(wh Av ζk , Sp Sq∗ ) =
Z
1
The rest of the inner products in (4.115) are directly integrated by the sinc quadrature
rule in space and time as
00
(wh A2v ζk , Sp Sq∗ )
∞
1
s
Υ0 (t)
=
dzdt
φ0 (z)
0
0
Z ∞
Z 1 00
p 0
wh (z)Sp (z)
2
∗
=
dz
A
(t)ζ
(t)S
(t)
Υ (t)dt
k
0
v
q
(φ (z))1/2
0
0
00
wh (zp ) A2v (tk )
p
,
≈ h z ht 0
(φ (zp ))3/2 Υ0 (tk )
Z
Z
00
wh (z)A2v (t)ζk (t)Sp (z)Sq∗ (t)
121
(wh (Av w∞ ) , Sp Sq∗ ) =
=
=
+
≈
and
00
(wh A2v w∞ , Sp Sq∗ )
∞
1
s
Υ0 (t)
dzdt
φ0 (z)
0
0
Z ∞
Z 1
p 0
wh (z)Sp (z)
0 ∗
(A
(t)w
(t))
S
(t)
dz
Υ (t)dt
v
∞
q
(φ0 (z))1/2
0
0
½Z ∞
Z 1
p 0
wh (z)Sp (z)
0
∗
(A
(t))
w
(t)S
(t)
Υ (t)dt
dz
v
∞
q
(φ0 (z))1/2
0
0
¾
Z ∞
p 0
0 ∗
(4.118)
Av (t))(w∞ (t)) Sq (t) Υ (t)dt
0
½
¾
0
w∞ (tq ) 0
wh (zp )
w∞ (tq )
p 0
Av (tq ) + p 0
h z ht 0
Av (tq ) ,
(φ (zp ))3/2
Υ (tq )
Υ (tq )
Z
0
Z
∞
0
wh (z)(Av (t)w∞ (t)) Sp (z)Sq∗ (t)
1
s
Υ0 (t)
dzdt
=
φ0 (z)
0
0
Z ∞
Z 1 00
p 0
wh (z)Sp (z)
2
∗
dz
A
(t)w
(t)S
(t)
=
Υ (t)dt
∞
v
q
(φ0 (z))1/2
0
0
00
wh (zp ) A2v (tq )w∞ (tq )
p 0
.
(4.119)
≈ h z ht 0
(φ (zp ))3/2
Υ (tq )
Z
Z
00
wh (z)A2v (t)w∞ (t)Sp (z)Sq∗ (t)
Therefore, (4.115) becomes
(Lwb , Sp Sq∗ )
≈
−
+
−
¾
½
Nt ·
X
p 0
1 (1) 1 (0)
wh (zp )
c
δ
A
(t
)
Υ (tk )
−
δ
+
h z ht σ
v k
0
qk
qk
3/2 (Nz +1)k
(φ
(z
h
2
p ))
t
k=−Mt
¸
00
A2v (tk )
wh (zp )
c(N +1)k p 0
(φ0 (zp ))3/2 z
Υ (tk )
·
½
¾
0
wh (zp )
w∞ (tq ) 0
w∞ (tq )
h z ht σ
c(N +1)(Nt +1) p 0
Av (tq ) + p 0
Av (tq )
(φ0 (zp ))3/2 z
Υ (tq )
Υ (tq )
¸
00
A2v (tq )w∞ (tq )
wh (zp )
c(N +1)(Nt +1) p 0
.
(4.120)
(φ0 (zp ))3/2 z
Υ (tq )
Letting p, q range over all values gives the matrix approximation to (Lwb , Sp Sq∗ ),
hz ht σ {Ab CDtb + Dsb CB b } ,
(4.121)
122
where the mz × mz block matrices
¯
¯
¤
fc £
¯
¯ ahN +1 ,
O
O
m
×1
m
×n
z
z
z
z
2
2κ
¯
¯
¤
£
≡
O mz ×1 ¯ O mz ×nz ¯ bhNz +1 ,
Ab ≡
D sb
(4.122)
the mt × mt block matrices
Dtb = Dt Dmt (Av ),

B Tw

Bb ≡ 

aT∞
(4.123)




 Dmt (Av ) + 


O nt ×mt
bT∞


 Dmt (A0v ),

and Dt is in (4.83). The mz × 1 column vectors ahNz +1 , bhNz +1 , have pth component,
−Mz − 1 ≤ p ≤ Nz + 1,
00
[ ahNz +1 ]p ≡ −
wh (zp )
wh (zp )
h
,
[
b
]
≡
,
p
0
N
+1
z
(φ (zp ))3/2
(φ0 (zp ))3/2
and the mt × 1 column vectors a∞ , b∞ , have q th component, −Mt ≤ q ≤ Nt + 1,
0
[ a∞
w (tq )
w∞ (tq )
]q ≡ p∞ 0
, [ b ∞ ]q ≡ p 0
.
Υ (tq )
Υ (tq )
In the Sinc-Galerkin method, the solution wa (z, t) in (4.109) is usually evaluated
at the nodal points (zp , tq ), where
zp =
eph
and tq = eqh .
ph
1+e
So we will construct the evaluator mz × mt matrix. First we start with
wa (zp , tq ) = wa (zp , tq ) + wb (zp , tq )
=
N
z +1
X
N
t +1
X
j=−Mz −1 k=−Mt
cjk ξj∗ (zp )ζk (tq )
(4.124)
123
N
t +1
X
+ σAv (tq )
c(Nt +1)k wh (zp )ζk (tq )
k=−Mt
=
£
c(−Mz −1)q w0 (zp ) + c(−Mz −1)(Nt +1) w0 (zp )w∞ (tq )
+ cpq
1
1
+ cp(Nt +1) 0
w∞ (tq )
φ (zp )
φ (zp )
0
+ c(Nz +1)q wg (zp ) + c(Nz +1)(Nt +1) wg (zp )w∞ (tq )
¤
£
¤
+ σAv (tq ) c(Nz +1)q wh (zp ) + c(Nz +1)(Nt +1 )wh (zp )w∞ (tq ) .
Letting p, q range over all values gives the mz × mt evaluator matrix approximation
to wa (zp , tq ),
{ Aea CDea + σAeb CDeb } ,
(4.125)
where the mz × mz block matrices
A ea ≡
A eb ≡
·
·
e0−Mz −1
¯ (0)
¯ Iz D n z
¯
O mz ×1 ¯ O mz ×nz
The mt × mt block matrices

Int ×nt
¶
¸
¯ g
1
¯
eNz +1 ,
φ0
¸
¯ h
¯ eN +1 .
z
µ
O nt ×1

D ea ≡ 

D eb
eT∞
≡ Dea Dmt (Av ),


,

and the mz × 1 column vectors with pth component, −Mz − 1 ≤ p ≤ Nz + 1,
£
e0−Mz −1
£
£
egNz +1
ehNz +1
¤
¤
¤
p
≡ w0 (zp ),
p
≡ wg (zp ),
p
≡ wh (zp ).
(4.126)
124
The mt × 1 column vectors with q th component, −Mt ≤ q ≤ Nt + 1,
[ e∞ ]q ≡ w∞ (tq ).
Next, applying the sinc quadrature rule, the inner product (wa , Sp Sq∗ ) is
(wa , Sp Sq∗ )
∞
1
s
Υ0 (t)
=
dzdt
φ0 (z)
0
0
1
1
wa (zp , tq ) p 0
,
≈ h z ht 0
3/2
(φ (zp ))
Υ (tq )
Z
Z
wa (z, t)Sp (z)Sq∗ (t)
(4.127)
where wa (zp , tq ) is given by (4.125). Substituting wa into (4.127) and letting p, q range
over all values gives the matrix approximation to (wa , Sp Sq∗ ),
hz ht { Ds CDt∗ + σDsb CDt } ,
(4.128)
¡
¢
where Dt is in (4.83), Ds is in (4.81), Dsb is in (4.122), and Dt∗ = Dt D 1/Av .
Using the expressions (4.112), (4.121), (4.128), and (4.79) substitute into (4.111),
this yields the discrete system
Ag CDt + Dg CB + σAb CDtb + σDsb CB b − (ifc k2 )Ds CDt∗ − (ifc k2 σ)Dsb CDt = F,
(4.129)
where Ag , Dg are in (4.81), F is in (4.82), B, Dt are in (4.83), Ab , Dsb are in (4.122),
and Dtb , Bb are in (4.123).
With Theorem 6 in Chapter 2, (4.129) is algebraically equivalent to the linear
discrete system
Gco(C) = co(F),
(4.130)
125
where the Kronecker product is given by a (mz mt ) × (mz mt ) matrix
G = DtT ⊗ Ag + B T ⊗ Dg + DtTb ⊗ σAb + BbT ⊗ σDsb
(4.131)
−DtT∗ ⊗ (ifc k2 )Ds − DtT ⊗ (ifc k2 )σDsb ,
and co(C) and co(F) are (mz mt ) × 1 vectors.
Numerical Examples for Time-Dependent Boundary Conditions
The examples in this section illustrate the approximate solution for problems
with time-dependent boundary conditions and address various features of different
sinc and uniform grids, the time-dependent eddy viscosities and the parameters.
Example 12. First consider an increasing time-dependent eddy viscosity Av (t) = 2 −
e−2t in Figure 42, with the parameters σ = 1, 2κ2 = 1, k2 = 0, and fc = 1. The
initial-boundary-value problem (4.97)-(4.100) which have time-dependent boundary
conditions becomes
µ
¶
∂w(z, t)
∂
−2t ∂w(z, t)
−
(2 − e )
= F (z, t), 0 < z < 1, 0 < t
∂t
∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
w(1, t) + (2 − e−2t )
= 0, 0 < t
∂z
w(z, 0) = 0, 0 < z < 1.
With the forcing function given by
¡
¢
F (z, t) = (z − z 2 )2 (1 − t)e−t − (2 − e−2t ) 2(1 − 2z)2 te−t − 4(z − z 2 )te−t ,
126
the exact real-valued solution is
w(z, t) = (z − z 2 )2 te−t .
0.3
0.5
2.2
0.7
Av (t)
2.0
-0.05
2
1.8
1.6
-0.1
1.4
-0.15
-0.2
1.2
T0 = .45
1
0
1
2
3
4
5
6
7
8
9
10
t
Figure 42. The graph of the time-dependent increasing eddy viscosity Av (t) = 2−e−2t .
The discrete system is given by (4.130) and we use the parameter choices α = β =
γ = 1 in (4.90) with Nt determined by experimentation. The results are reported in
Table 14, Table 15, and Figure 43, Figure 44, Figure 45, and Figure 46, respectively.
Table 14 again suggests that we choose Nt = (1/2)Mz . Table 15 shows the errors
with Mz = Nz = Mt = 2Nt for Mz = 4, 8, 16, 32. With time-dependent boundary
conditions, the results are slightly better than those of Example 11 which had timeindependent boundary conditions. The approximate solution is graphed in Figure 43
using the values generated on the sinc grid S.
The approximate solution wa (z, t)
is graphed in Figure 44 using the values on a uniform grid U .
The approximate
solution is graphed in Figure 45 using the values generated on the sinc grid S t . This
127
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 7.197e-03
1.111 7.196e-03
1.111 7.200e-03
1.111 7.199e-03
0.785 1.776e-03
0.785 1.696e-03
0.785 1.695e-03
0.785 1.695e-03
0.785 1.695e-03
0.785 1.695e-03
0.785 1.695e-03
0.785 1.695e-03
0.555 4.642e-04
0.555 1.923e-04
0.555 1.614e-04
0.555 1.608e-04
0.555 1.607e-04
0.555 1.608e-04
0.555 1.607e-04
0.555 1.608e-04
0.555 1.607e-04
0.555 1.607e-04
0.555 1.607e-04
0.555 1.607e-04
0.555 1.607e-04
0.555 1.607e-04
0.555 1.607e-04
0.555 1.607e-04
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
7.197e-03
7.196e-03
7.200e-03
7.199e-03
1.776e-03
1.696e-03
1.695e-03
1.695e-03
1.695e-03
1.695e-03
1.695e-03
1.695e-03
4.642e-04
1.923e-04
1.614e-04
1.608e-04
1.607e-04
1.608e-04
1.607e-04
1.608e-04
1.607e-04
1.607e-04
1.607e-04
1.607e-04
1.607e-04
1.607e-04
1.607e-04
1.607e-04
8.295e-03
8.292e-03
8.309e-03
8.311e-03
2.826e-03
1.706e-03
1.703e-03
1.703e-03
1.703e-03
1.703e-03
1.703e-03
1.703e-03
1.071e-02
1.871e-03
1.662e-04
1.664e-04
1.664e-04
1.665e-04
1.666e-04
1.666e-04
1.666e-04
1.667e-04
1.667e-04
1.667e-04
1.667e-04
1.667e-04
1.667e-04
1.667e-04
Table 14. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 12 with Av (t) = 2 − e−2t and σ = 1.
128
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
kVS k
kES k
1.111 7.196e-03 0.000e+00
0.785 1.695e-03 0.000e+00
0.555 1.608e-04 0.000e+00
0.393 5.132e-06 0.000e+00
kEU k
7.196e-03 8.292e-03
1.695e-03 1.703e-03
1.608e-04 1.666e-04
5.132e-06 5.155e-06
Table 15. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 12 with Av (t) = 2 − e−2t and σ = 1.
PSfrag replacements
0
0.025
wa (z, t)
0.02
0.015
0.01
0.005
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
t
2
0
0
Figure 43. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 12 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
0
129
0.025
0.02
wa (z, t)
0.015
0.01
0.005
0
1
0.8
10
0.6
8
z
6
0.4
4
0.2
PSfrag replacements
t
2
0
0
Figure 44. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 12 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
0.025
z=0
z = .16
z = .50
z=1
0.02
wa (z, t)
0.015
0.01
0.005
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 45. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
for Example 12 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
130
time plot is graphed for z = 0, .16, .50, and 1. The approximate solution is graphed
in Figure 46 using the values on a uniform grid U t . This time plot is graphed for
z = 0, .20, .50, and 1.
0.025
z=0
z = .20
z = .50
z=1
0.02
wa (z, t)
0.015
0.01
0.005
0
0
1
2
3
4
5
t
6
7
8
9
10
Figure 46. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 12 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt =
16, Nt = 8.
Example 13. From Example 12 we use the same time-dependent eddy viscosity Av (t) =
2 − e−2t in Figure 42 and the parameters σ = 1, 2κ2 = 1, fc = 1, and k2 = 1. So the
full model for the initial-boundary-value problem (4.97)-(4.100) becomes
µ
¶
∂
∂w(z, t)
−2t ∂w(z, t)
−
(2 − e )
− iw(z, t) = F (z, t), 0 < z < 1, 0 < t
∂t
∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
= 0, 0 < t
w(1, t) + (2 − e−2t )
∂z
w(z, 0) = 0, 0 < z < 1.
131
With the forcing function given by
¢
¡
F (z, t) = (z − z 2 )2 (1 − t)e−t − (2 − e−2t ) 2(1 − 2z)2 te−t − 4(z − z 2 )te−t − i(z − z 2 )2 te−t ,
the exact real-valued solution is
w(z, t) = (z − z 2 )2 te−t .
Note that the solution is real, so the imaginary part is zero. The discrete system
is given by (4.130) and we use the parameter choices α = β = γ = 1 in (4.90).
The results are reported in Table 16, Table 17 and Figure 47, Figure 48, Figure 49,
and Figure 50, respectively. Table 16 also suggests that we choose Nt = (1/2)Mz .
Table 17 shows the errors with Mz = Nz = Mt = 2Nt for Mz = 4, 8, 16, 32.
The results are slightly better than those of Example 12 which had k2 = 0.
The
approximate solution is graphed in Figure 47 using the values generated on the sinc
grid S. The approximate solution is graphed in Figure 48 using the values generated
on the uniform grid U . The approximate solution is graphed in Figure 49 using the
values on the sinc grid S t . This time plot is graphed for z = 0, .16, .50 and 1. The
approximate solution is graphed in Figure 50 using the values on the uniform grid
U t . This time plot is graphed for z = 0, .20, .50 and 1.
132
Mz
Nz
Mt
Nt
h
kUS k
kVS k
kES k
kEU k
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1.111
1.111
1.111
1.111
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
5.541e-03
5.525e-03
5.528e-03
5.527e-03
1.101e-03
1.067e-03
1.068e-03
1.067e-03
1.067e-03
1.067e-03
1.067e-03
1.067e-03
4.439e-04
1.093e-04
1.277e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
4.161e-03
4.165e-03
4.169e-03
4.168e-03
1.012e-03
9.725e-04
9.712e-04
9.711e-04
9.712e-04
9.711e-04
9.712e-04
9.711e-04
2.004e-04
1.127e-04
8.874e-05
8.914e-05
8.915e-05
8.917e-05
8.917e-05
8.918e-05
8.918e-05
8.918e-05
8.918e-05
8.918e-05
8.918e-05
8.918e-05
8.918e-05
8.918e-05
5.541e-03
5.525e-03
5.528e-03
5.527e-03
1.101e-03
1.067e-03
1.068e-03
1.067e-03
1.067e-03
1.067e-03
1.067e-03
1.067e-03
4.439e-04
1.127e-04
1.277e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
1.273e-04
5.907e-03
5.925e-03
5.936e-03
5.938e-03
2.471e-03
1.286e-03
1.286e-03
1.286e-03
1.287e-03
1.287e-03
1.287e-03
1.287e-03
1.070e-02
1.854e-03
1.284e-04
1.279e-04
1.279e-04
1.279e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
1.278e-04
Table 16. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 13 with Av (t) = 2 − e−2t and σ = 1.
133
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
1.111 5.525e-03
0.785 1.067e-03
0.555 1.273e-04
0.393 3.711e-06
kVS k
kES k
kEU k
4.165e-03 5.525e-03 5.925e-03
9.711e-04 1.067e-03 1.286e-03
8.918e-05 1.273e-04 1.278e-04
2.912e-06 3.711e-06 3.842e-06
Table 17. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 13 with Av (t) = 2 − e−2t and σ = 1.
PSfrag replacements
0
0.025
wa (z, t)
0.02
0.015
0.01
0.005
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
t
2
0
0
Figure 47. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 13 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
134
0
0.025
0.02
wa (z, t)
0.015
0.01
0.005
0
1
0.8
PSfrag replacements
10
0.6
z
8
6
0.4
4
0.2
t
2
0
0
Figure 48. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 13 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
0.025
z=0
z = .16
z = .50
z=1
0.02
wa (z, t)
0.015
0.01
0.005
0
0
1
2
3
4
5
t
6
7
8
9
10
Figure 49. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
for Example 13 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
135
0.025
z=0
z = .20
z = .50
z=1
0.02
wa (z, t)
0.015
0.01
0.005
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 50. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 13 where Av (t) = 2 − e−2t and σ = 1 with Mz = Nz = Mt =
16, Nt = 8.
Example 14. Choosing a different time-dependent eddy viscosity Av (t) = 1 + te1−t
in Figure 51 which is similar to a realistic eddy viscosity and the parameters σ = 1,
2κ2 = 1, k2 = 0 and fc = 1. The initial-boundary-value problem (4.97)-(4.100)
becomes
µ
¶
∂w(z, t)
∂
1−t ∂w(z, t)
−
(1 + te )
= F (z, t), 0 < z < 1, 0 < t
∂t
∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
w(1, t) + (1 + te1−t )
= 0, 0 < t
∂z
w(z, 0) = 0, 0 < z < 1.
136
With the forcing function given by
5
(z − z 2 )7/2 t3/2 e−t − (z − z 2 )7/2 t5/2 e−t
2
µ
¶
35
1−t
2 3/2 5/2 −t
2
2 5/2 5/2 −t
− (1 + te )
,
(z − z ) t e (1 − 2x) − 7(z − z ) t e
4
F (z, t) =
the exact real-valued solution is
w(z, t) = (z − z 2 )7/2 t5/2 e−t .
The function Av (t) in Figure 51 is chosen for its similarity to an eddy viscosity
function used in Chapter 5. Note Av (t) increases to its maximum value at t = 1 and
then decreases thereafter. The discrete system is given by (4.130) and we use the
parameter choices α = β = γ = 1 in (4.90). The results are reported in Table 18,
Table 19, and Figure 52, Figure 53, Figure 54, and Figure 55, respectively. Table
18 again suggests that we choose Nt = (1/2)Mz
Table 19 shows the errors with
Mz = Nz = Mt = 2Nt for Mz = 4, 8, 16, 32. The results are better than those of
Example 12 which had the increasing eddy viscosity.
The approximate solution is
graphed in Figure 52 using the values generated on the sinc grid S.
The approximate solution wa (z, t) is graphed in Figure 53 using the values on
the uniform grid U .
The approximate solution is graphed in Figure 54 using the
values on the sinc grid S t . This time plot is graphed for z = 0, .16, .50 and 1. The
approximate solution is graphed in Figure 55 using the values on the uniform grid
U t . This time plot is graphed for z = 0, .20, .50 and 1.
137
2.2
0.3
2
0.5
1.8
Av (t)
0.7
2.0
-0.05
1.6
1.4
-0.1
-0.15
1.2
-0.2
T0 = .45
1
0
1
2
4
3
5
6
7
8
9
10
t
PSfrag replacements
Figure 51. The graph of the time-dependent eddy viscosity Av (t) = 1 + te1−t with a
zero steady-state.
2
4
6
−3
x 10
7
6
5
wa (z, t)
1
4
3
2
1
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 52. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 14 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
138
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 3.588e-03
1.111 3.354e-03
1.111 3.393e-03
1.111 3.376e-03
0.785 5.409e-04
0.785 4.296e-04
0.785 4.335e-04
0.785 4.279e-04
0.785 4.309e-04
0.785 4.289e-04
0.785 4.301e-04
0.785 4.294e-04
0.555 2.376e-04
0.555 1.345e-04
0.555 2.195e-05
0.555 2.469e-05
0.555 2.241e-05
0.555 2.383e-05
0.555 2.274e-05
0.555 2.346e-05
0.555 2.294e-05
0.555 2.330e-05
0.555 2.304e-05
0.555 2.322e-05
0.555 2.310e-05
0.555 2.319e-05
0.555 2.312e-05
0.555 2.317e-05
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
3.588e-03
3.354e-03
3.393e-03
3.376e-03
5.409e-04
4.296e-04
4.335e-04
4.279e-04
4.309e-04
4.289e-04
4.301e-04
4.294e-04
2.376e-04
1.345e-04
2.195e-05
2.469e-05
2.241e-05
2.383e-05
2.274e-05
2.346e-05
2.294e-05
2.330e-05
2.304e-05
2.322e-05
2.310e-05
2.319e-05
2.312e-05
2.317e-05
3.588e-03
3.409e-03
3.464e-03
3.452e-03
3.506e-03
5.018e-04
5.131e-04
5.073e-04
5.146e-04
5.118e-04
5.144e-04
5.132e-04
6.871e-03
2.628e-03
3.447e-05
4.808e-05
4.585e-05
4.640e-05
4.544e-05
4.579e-05
4.536e-05
4.555e-05
4.535e-05
4.545e-05
4.535e-05
4.541e-05
4.536e-05
4.538e-05
Table 18. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 14 with Av (t) = 1 + te1−t and σ = 1.
139
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
kVS k
kES k
1.111 3.354e-03 0.000e+00
0.785 4.279e-04 0.000e+00
0.555 2.346e-05 0.000e+00
0.393 6.458e-07 0.000e+00
kEU k
3.354e-03 3.409e-03
4.279e-04 5.073e-04
2.346e-05 4.579e-05
6.458e-07 1.537e-06
Table 19. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 14 with Av (t) = 1 + te1−t and σ = 1.
PSfrag replacements
2
4
6
−3
x 10
7
6
1
wa (z, t)
5
4
3
2
1
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 53. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 14 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
140
7
x 10
−3
z=0
z = .16
z = .50
z=1
6
5
wa (z, t)
4
3
2
1
0
0
1
2
4
3
5
6
t
8
7
9
10
54. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
PSfrag Figure
replacements
for Example 14 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
−3
7
x 10
z
z
z
z
6
=0
= .20
= .50
=1
5
wa (z, t)
4
3
2
1
0
-1
0
1
2
3
4
5
6
7
8
9
10
t
Figure 55. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 14 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt =
16, Nt = 8.
141
Example 15. From Example 14, we use the same time-dependent eddy viscosity
Av (t) = 1 + te1−t in Figure 51 and the parameter choices σ = 1, 2κ2 = 1, and
fc = 1. Choosing k2 = 1, the full model for the initial-boundary-value problem
(4.97)-(4.100) becomes
µ
¶
∂
∂w(z, t)
1−t ∂w(z, t)
−
(1 + te )
− iw(z, t) = F (z, t), 0 < z < 1, 0 < t
∂t
∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
= 0, 0 < t
w(1, t) + (1 + te1−t )
∂z
w(z, 0) = 0, 0 < z < 1.
With the forcing function given by
5
(z − z 2 )7/2 t3/2 e−t − (z − z 2 )7/2 t5/2 e−t
2
¶
µ
35
2 3/2 5/2 −t
2
2 5/2 5/2 −t
1−t
(z − z ) t e (1 − 2z) − 7(z − z ) t e
− (1 + te )
4
F (z, t) =
− i(z − z 2 )7/2 t5/2 e−t ,
the exact real-values solution is
w(z, t) = (z − z 2 )7/2 t5/2 e−t .
The discrete system is given by (4.130) and we use the parameter choices α = β =
γ = 1 in (4.90). The results are reported in Table 20, Table 21, and Figure 56,
Figure 57, Figure 58, and Figure 59, respectively. Table 20 again suggests that we
choose Nt = (1/2)Mz . Table 21 shows the errors with Mz = Nz = Mt = 2Nt for
Mz = 4, 8, 16, 32. The results are very similar to those of Example 14 which had
142
k2 = 0. The approximate solution is graphed in Figure 56 using the values generated
PSfrag replacements
on the sinc grid S. The approximate solution is graphed in Figure 57 using the values
on the2 uniform grid U .
The approximate solution is graphed in Figure 58 using
4
6
10
-3
x 10
7
6
1
wa (z, t)
5
4
3
2
1
×
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 56. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 15 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
the values on the sinc grid S t . This time plot is graphed for z = 0, .16, .50 and 1.
The approximate solution is graphed in Figure 59 using the values on the uniform
grid U t . This time plot is graphed for z = 0, .20, .50 and 1.
143
Mz
Nz
Mt
Nt
h
kUS k
kVS k
kES k
kEU k
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1.111
1.111
1.111
1.111
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
1.872e-03
1.918e-03
1.909e-03
1.913e-03
3.918e-04
3.477e-04
3.500e-04
3.488e-04
3.494e-04
3.491e-04
3.493e-04
3.492e-04
1.857e-04
8.848e-05
1.389e-05
1.624e-05
1.591e-05
1.611e-05
1.598e-05
1.606e-05
1.601e-05
1.604e-05
1.602e-05
1.603e-05
1.602e-05
1.603e-05
1.602e-05
1.602e-05
1.858e-03
1.788e-03
1.800e-03
1.796e-03
2.502e-04
1.940e-04
2.028e-04
1.978e-04
2.012e-04
1.993e-04
2.004e-04
1.997e-04
9.717e-05
6.097e-05
1.167e-05
1.326e-05
1.160e-05
1.266e-05
1.188e-05
1.239e-05
1.203e-05
1.227e-05
1.210e-05
1.222e-05
1.214e-05
1.220e-05
1.215e-05
1.218e-05
1.872e-03
1.918e-03
1.909e-03
1.913e-03
3.918e-04
3.477e-04
3.500e-04
3.488e-04
3.494e-04
3.491e-04
3.493e-04
3.492e-04
1.857e-04
8.848e-05
1.389e-05
1.624e-05
1.591e-05
1.611e-05
1.598e-05
1.606e-05
1.601e-05
1.604e-05
1.602e-05
1.603e-05
1.602e-05
1.603e-05
1.602e-05
1.602e-05
1.981e-03
2.007e-03
1.999e-03
2.003e-03
3.292e-03
4.577e-04
4.613e-04
4.586e-04
4.588e-04
4.580e-04
4.581e-04
4.579e-04
6.928e-03
2.672e-03
3.310e-05
4.515e-05
4.425e-05
4.441e-05
4.405e-05
4.414e-05
4.399e-05
4.404e-05
4.397e-05
4.400e-05
4.397e-05
4.398e-05
4.397e-05
4.397e-05
Table 20. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 15 with Av (t) = 1 + te1−t and σ = 1.
144
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
1.111 1.918e-03
0.785 3.488e-04
0.555 1.606e-05
0.393 4.816e-07
kVS k
kES k
kEU k
1.788e-03 1.918e-03 2.007e-03
1.978e-04 3.488e-04 4.586e-04
1.239e-05 1.606e-05 4.414e-05
3.060e-07 4.816e-07 1.654e-06
Table 21. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 15 with Av (t) = 1 + te1−t and σ = 1.
PSfrag replacements
2
4
6
−3
x 10
7
6
1
wa (z, t)
5
4
3
2
1
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 57. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 15 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
145
7
x 10
−3
z=0
z = .16
z = .50
z=1
6
5
wa (z, t)
4
3
2
1
0
0
1
2
4
3
5
6
t
8
7
9
10
Figure 58. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
Example 15 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
PSfragfor
replacements
−3
7
x 10
z
z
z
z
6
=0
= .20
= .50
=1
5
wa (z, t)
4
3
2
1
0
−1
0
1
2
3
4
5
6
7
8
9
10
t
Figure 59. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 15 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt =
16, Nt = 8.
146
The next several numerical examples test nonzero steady-state problems with
known solutions. They are given to illustrate the performance of the Sinc-Galerkin
method.
Example 16. First consider a time-dependent eddy viscosity Av (t) = 1 + te1−t in
Figure 51, with the parameters σ = 1, 2κ2 = 1, fc = 1, and k2 = 0. The initialboundary-value problem (4.97)-(4.100) becomes
µ
¶
∂w(z, t)
∂
1−t ∂w(z, t)
(1 + te )
= F (z, t), 0 < z < 1, 0 < t
−
∂t
∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
w(1, t) + (1 + te1−t )
= 0, 0 < t
∂z
w(z, 0) = 0, 0 < z < 1.
With the forcing function
¡
F (z, t) = (z(1 − z))2 e−t − (1 + te1−t ) (2 − 12z + 12z 2 )(1 − e−t )
the exact real-valued solution is
w(z, t) = (z(1 − z))2 (1 − e1−t ).
The discrete system is given by (4.130) and we use the parameter choices α = β =
γ = 1 in (4.90). The results are reported in Table 22, Table 23, and Figure 60,
Figure 61, Figure 62, and Figure 63, respectively. Table 22 again suggests that we
choose Nt = (1/2)Mz . Table 23 shows the errors with Mz = Nz = Mt = 2Nt for
Mz = 4, 8, 16, 32. For the nonzero steady-state, the results are the same as for
147
examples which have a zero steady-state. The approximate solution is graphed in
Figure 60 using the values generated on the sinc grid S. The approximate solution
is graphed in Figure 61 using the values on the uniform grid U . The approximate
solution is graphed in Figure 62 using the values on the sinc grid S t . This time plot is
graphed for z = 0, .16, .50 and 1. The approximate solution is graphed in Figure 63
using the values on the uniform grid U t . This time plot is graphed for z = 0, .20, .50
and 1.
PSfrag replacements
0.07
0.06
wa (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 60. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 16 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
148
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 2.595e-02
1.111 2.521e-02
1.111 2.523e-02
1.111 2.516e-02
0.785 5.087e-03
0.785 5.335e-03
0.785 5.233e-03
0.785 5.229e-03
0.785 5.222e-03
0.785 5.218e-03
0.785 5.218e-03
0.785 5.216e-03
0.555 4.400e-04
0.555 5.739e-04
0.555 5.301e-04
0.555 4.837e-04
0.555 4.868e-04
0.555 4.817e-04
0.555 4.837e-04
0.555 4.831e-04
0.555 4.831e-04
0.555 4.830e-04
0.555 4.830e-04
0.555 4.829e-04
0.555 4.829e-04
0.555 4.829e-04
0.555 4.829e-04
0.555 4.829e-04
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
2.595e-02
2.521e-02
2.523e-02
2.516e-02
5.087e-03
5.335e-03
5.233e-03
5.229e-03
5.222e-03
5.218e-03
5.218e-03
5.216e-03
4.400e-04
5.739e-04
5.301e-04
4.837e-04
4.868e-04
4.817e-04
4.837e-04
4.831e-04
4.831e-04
4.830e-04
4.830e-04
4.829e-04
4.829e-04
4.829e-04
4.829e-04
4.829e-04
2.710e-02
2.678e-02
2.679e-02
2.674e-02
1.009e-02
5.404e-03
5.290e-03
5.290e-03
5.282e-03
5.279e-03
5.278e-03
5.277e-03
9.480e-03
5.028e-03
6.030e-04
4.929e-04
5.028e-04
4.996e-04
5.010e-04
5.005e-04
5.005e-04
5.004e-04
5.003e-04
5.003e-04
5.003e-04
5.003e-04
5.003e-04
5.003e-04
Table 22. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 16 with Av (t) = 1 + te1−t and σ = 1.
149
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
kVS k
kES k
1.111 2.521e-02 0.000e+00
0.785 5.229e-03 0.000e+00
0.555 4.831e-04 0.000e+00
0.393 1.515e-05 0.000e+00
kEU k
2.521e-02 2.678e-02
5.229e-03 5.290e-03
4.831e-04 5.005e-04
1.515e-05 1.540e-05
Table 23. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 16 with Av (t) = 1 + te1−t and σ = 1.
PSfrag replacements
0.07
0.06
wa (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
t
2
0
0
Figure 61. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 16 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
150
0.2
0.4
0.6
0.07
0.8
0.06
PSfrag replacements
wa (z, t)
0.05
z
z
z
z
0.04
=0
= .16
= .50
=1
0.03
1
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 62. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
for Example 16 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
0.2
0.4
0.6
0.07
0.8
0.06
wa (z, t)
0.05
0.04
0.03
0.02
z
z
z
z
0.01
0
0
1
2
3
4
5
6
7
=0
= .20
= .50
=1
8
9
10
t
Figure 63. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 16 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt =
16, Nt = 8.
151
The following numerical examples test for known complex-valued solutions. They
are given to illustrate the performance of the Sinc-Galerkin method for a known
complex-valued solution.
Example 17. From Example 16 with the same time-dependent eddy viscosity Av (t) =
1 + te1−t in Figure 51, we choose k2 = 1, and the full model for the initial-boundaryvalue problem (4.97)-(4.100) becomes
µ
¶
∂w(z, t)
fc ∂
1−t ∂w(z, t)
(1 + te )
− ik2 fc w(z, t) = F (z, t), 0 < z < 1, 0 < t
− 2
∂t
2κ ∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
w(1, t) + σ(1 + te1−t )
= 0, 0 < t
∂z
w(z, 0) = 0, 0 < z < 1,
where we select σ = 0.1, κ = 2.35, and fc = 4.95. With the forcing function given by
F (z, t) =
¡
¢
z 2 (1 − z)2 + iz 2 (1 − z)2 e−t
¡
¢
fc
1−t
2
2
(1
+
te
)
(2
−
12z
+
12z
)
+
i(2
−
12z
+
12z
)
(1 − e−t )
2κ2
¡
¢
− ik2 fc z 2 (1 − z)2 + iz 2 (1 − z)2 (1 − e1−t ),
−
the exact complex-valued solution is
¡
¢
w(z, t) = z 2 (1 − z)2 + iz 2 (1 − z)2 (1 − e−t ).
The discrete system is given by (4.130) and we use the parameter choices α = β =
γ = 1 in (4.90). The results are reported in Table 24, Table 25, and Figure 64,
152
Figure 66, Figure 68, and Figure 70, respectively. Table 24 again suggests that we
choose Nt = (1/2)Mz . Table 25 shows the errors with Mz = Nz = Mt = 2Nt for
Mz = 4, 8, 16, 32. The results are similar to those of Example 16. Both real and
imaginary parts of the approximate solution are graphed in Figure 64 and Figure 65
by using the values generated on the sinc grid S. The real and imaginary parts of
the approximate solution wa (z, t) are also graphed in Figure 66 and Figure 67 by
using the values on the uniform grid U . The time plots are graphed in Figure 68 and
Figure 69 for z = 0, .16, .50 and 1 by using the values on the uniform grid S t . The
PSfrag replacements
time plots of the approximate solution are graphed in Figure 70 and Figure 71 for
z = 0, .20, .50 and 1 by using the values on the uniform grid U t .
0.07
0.06
ua (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 64. The graph of the approximate solution (real part) ua (z, t) on the sinc grid
S for Example 17 with Av (t) = 1 + te1−t with Mz = Nz = Mt = 16, Nt = 8.
153
Mz
Nz
Mt
Nt
h
kUS k
kVS k
kES k
kEU k
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1.111
1.111
1.111
1.111
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
2.860e-03
2.838e-03
2.844e-03
2.843e-03
6.628e-04
6.290e-04
6.040e-04
6.063e-04
6.056e-04
6.058e-04
6.058e-04
6.058e-04
1.495e-04
1.341e-04
9.003e-05
6.856e-05
6.100e-05
5.788e-05
5.707e-05
5.672e-05
5.678e-05
5.676e-05
5.676e-05
5.676e-05
5.676e-05
5.676e-05
5.676e-05
5.676e-05
9.879e-03
9.939e-03
9.937e-03
9.939e-03
2.249e-03
2.055e-03
2.094e-03
2.084e-03
2.086e-03
2.085e-03
2.086e-03
2.085e-03
4.212e-04
2.978e-04
2.387e-04
1.951e-04
2.094e-04
2.044e-04
2.062e-04
2.055e-04
2.058e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
9.879e-03
9.939e-03
9.937e-03
9.939e-03
2.249e-03
2.055e-03
2.094e-03
2.084e-03
2.086e-03
2.085e-03
2.086e-03
2.085e-03
4.212e-04
2.978e-04
2.387e-04
1.951e-04
2.094e-04
2.044e-04
2.062e-04
2.055e-04
2.058e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
2.057e-04
1.114e-02
1.108e-02
1.109e-02
1.109e-02
6.048e-03
2.190e-03
2.220e-03
2.209e-03
2.211e-03
2.210e-03
2.210e-03
2.210e-03
9.424e-03
4.902e-03
3.013e-04
2.004e-04
2.116e-04
2.054e-04
2.076e-04
2.068e-04
2.070e-04
2.069e-04
2.070e-04
2.070e-04
2.070e-04
2.070e-04
2.070e-04
2.070e-04
Table 24. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 17 with Av (t) = 1 + te1−t and σ = 0.1 .
154
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
1.111 2.838e-03
0.785 6.063e-04
0.555 5.672e-05
0.393 1.746e-06
kVS k
kES k
kEU k
9.939e-03 9.939e-03 1.108e-02
2.084e-03 2.084e-03 2.209e-03
2.055e-04 2.055e-04 2.068e-04
6.312e-06 6.312e-06 6.337e-06
Table 25. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt = 2Nt
for Example 17 with Av (t) = 1 + te1−t .
PSfrag replacements
0.07
0.06
va (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 65. The graph of the approximate solution (imaginary part) va (z, t) on the
sinc grid S for Example 17 with Av (t) = 1 + te1−t with Mz = Nz = Mt = 16, Nt = 8.
155
0.07
0.06
ua (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
PSfrag replacements
2
0
t
0
Figure 66. The graph of the approximate solution (real part) ua (z, t) on the uniform
grid U for Example 17 with Av (t) = 1 + te1−t with Mz = Nz = Mt = 16, Nt = 8.
0.07
0.06
va (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 67. The graph of the approximate solution (imaginary part) va (z, t) on the
uniform grid U for Example 17 with Av (t) = 1 + te1−t with Mz = Nz = Mt =
16, Nt = 8.
156
0.2
0.4
0.6
0.07
0.8
0.06
PSfrag replacements
ua (z, t)
0.05
z
z
z
z
0.04
=0
= .16
= .50
=1
0.03
1
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 68. The approximate solution (real part) ua (z, t) at z = 0, .16, .50, 1 on the
sinc grid S t for Example 17 where Av (t) = 1+te1−t with Mz = Nz = Mt = 16, Nt = 8.
0.2
0.4
0.6
0.07
0.8
0.06
va (z, t)
0.05
z
z
z
z
0.04
=0
= .16
= .50
=1
0.03
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 69. The approximate solution (imaginary part) va (z, t) at z = 0, .16, .50, 1
on the sinc grid S t for Example 17 where Av (t) = 1 + te1−t with Mz = Nz = Mt =
16, Nt = 8.
0.2
157
0.4
0.6
0.07
0.8
0.06
ua (z, t)
0.05
0.04
PSfrag replacements
0.03
1
0.02
z
z
z
z
0.01
0
0
1
2
3
4
5
6
7
=0
= .20
= .50
=1
8
9
10
t
Figure 70. The approximate solution (real part) ua (z, t) at z = 0, .20, .50, 1 on the
uniform grid U t for Example 17 where Av (t) = 1 + te1−t with Mz = Nz = Mt =
16, Nt = 8. 0.2
0.4
0.6
0.07
0.8
0.06
va (z, t)
0.05
0.04
0.03
0.02
z
z
z
z
0.01
0
0
1
2
3
4
5
6
7
=0
= .20
= .50
=1
8
9
10
t
Figure 71. The approximate solution (imaginary part) va (z, t) at z = 0, .20, .50, 1
on the uniform grid U t for Example 17 where Av (t) = 1 + te1−t with Mz = Nz =
Mt = 16, Nt = 8.
158
Example 18. Consider a time-dependent eddy viscosity Av (t) = 1 + te1−t , with the
parameter σ = 1, 2κ2 = 1, fc = 1, and k2 = 0. The initial-boundary-value problem
(4.97)-(4.100) becomes
µ
¶
∂w(z, t)
∂
1−t ∂w(z, t)
(1 + te )
= F (z, t), 0 < z < 1, 0 < t
−
∂t
∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
= 0, 0 < t
w(1, t) + (1 + te1−t )
∂z
w(z, 0) = 0, 0 < z < 1.
With the forcing function given by
1
1
F (z, t) = (3z 2 − 2z 3 − 1)e−t − (1 + te1−t )(6 − 12z)(1 − e−t ),
6
6
the exact real-valued solution is
1
w(z, t) = (3z 2 − 2z 3 − 1)(1 − e−t ).
6
The discrete system is given by (4.130) and we use the parameter choices α = β =
γ = 1 in (4.90). The results are reported in Table 26, Table 27, and Figure 72,
Figure 73, Figure 74, and Figure 75, respectively. Table 26 again suggests that we
choose Nt = (1/2)Mz . Table 27 shows the errors with Mz = Nz = Mt = 2Nt for
Mz = 4, 8, 16, 32. The approximate solution is graphed in Figure 72 using the
values generated on the sinc grid S. The approximate solution is graphed in Figure
73 using the values on the uniform grid U . The time plot of wa (z, t) is graphed for
z = 0, .16, .50 and 1 in Figure 74 using the values on the sinc grid S t and graphed
for z = 0, .20, .50 and 1 in Figure 75 using the values on the uniform grid U t .
159
Mz
Nz
Mt
Nt
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
h
kUS k
1.111 2.042e-04
1.111 3.325e-04
1.111 3.098e-04
1.111 3.097e-04
0.785 8.747e-04
0.785 2.912e-04
0.785 3.803e-05
0.785 4.443e-05
0.785 3.623e-05
0.785 3.812e-05
0.785 3.756e-05
0.785 3.777e-05
0.555 8.900e-04
0.555 7.885e-04
0.555 3.409e-04
0.555 1.310e-04
0.555 5.183e-05
0.555 1.715e-05
0.555 6.997e-06
0.555 2.452e-06
0.555 3.146e-06
0.555 2.870e-06
0.555 2.964e-06
0.555 2.924e-06
0.555 2.940e-06
0.555 2.932e-06
0.555 2.936e-06
0.555 2.934e-06
kVS k
kES k
kEU k
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
0.000e+00
2.042e-04
3.325e-04
3.098e-04
3.097e-04
8.747e-04
2.912e-04
3.803e-05
4.443e-05
3.623e-05
3.812e-05
3.756e-05
3.777e-05
8.900e-04
7.885e-04
3.409e-04
1.310e-04
5.183e-05
1.715e-05
6.997e-06
2.452e-06
3.146e-06
2.870e-06
2.964e-06
2.924e-06
2.940e-06
2.932e-06
2.936e-06
2.934e-06
8.016e-04
9.508e-04
6.494e-04
7.186e-04
1.361e-02
9.714e-04
2.863e-04
5.677e-05
1.097e-04
8.598e-05
9.373e-05
9.071e-05
2.384e-02
1.212e-02
8.946e-04
2.556e-04
1.030e-04
4.104e-05
1.893e-05
1.072e-05
4.973e-06
5.581e-06
3.255e-06
4.404e-06
3.791e-06
4.109e-06
3.937e-06
4.029e-06
Table 26. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 18 with Av (t) = 1 + te1−t and σ = 1.
160
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
kUS k
h
kVS k
kES k
1.111 3.325e-04 0.000e+00
0.785 4.443e-05 0.000e+00
0.555 2.452e-06 0.000e+00
0.393 9.431e-08 0.000e+00
kEU k
3.325e-04 9.508e-04
4.443e-05 5.677e-05
2.452e-06 1.072e-05
9.431e-08 4.066e-07
Table 27. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 18 with Av (t) = 1 + te1−t and σ = 1.
PSfrag replacements
0
wa (z, t)
-0.05
-0.1
-0.15
-0.2
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 72. The graph of the approximate solution wa (z, t) on the sinc grid S for
Example 18 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
161
PSfrag replacements
0
wa (z, t)
-0.05
-0.1
PSfrag replacements
-0.15
-0.2
1
0.8
10
0.6
z
8
6
0.4
4
0.2
t
2
0
0
Figure 73. The graph of the approximate solution wa (z, t) on the uniform grid U for
Example 18 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
0
z
z
z
z
-0.02
=0
= .16
= .50
=1
-0.04
-0.06
wa (z, t)
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
0
1
2
3
4
5
t
6
7
8
9
10
Figure 74. The approximate solution wa (z, t) at z = 0, .16, .50, 1 on the sinc grid S t
for Example 18 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt = 16, Nt = 8.
162
0
z
z
z
z
-0.02
=0
= .20
= .50
=1
-0.04
-0.06
wa (z, t)
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
0
1
2
3
4
5
6
7
8
9
10
t
Figure 75. The approximate solution wa (z, t) at z = 0, .20, .50, 1 on the uniform
grid U t for Example 18 where Av (t) = 1 + te1−t and σ = 1 with Mz = Nz = Mt =
16, Nt = 8.
Example 19. The initial-boundary-value problem (4.97)-(4.100) with A v (t) = 1 +
3te1−t and σ = 1 is
µ
¶
∂w(z, t)
fc ∂
1−t ∂w(z, t)
− 2
(1 + 3te )
− ik2 fc w(z, t) = F (z, t), 0 < z < 1, 0 < t
∂t
2κ ∂z
∂z
∂w(0, t)
= 0, 0 < t
∂z
∂w(1, t)
w(1, t) + σ(1 + 3te1−t )
= 0, 0 < t
∂z
w(z, 0) = 0, 0 < z < 1,
where we select k2 = 1, σ = 0.1, κ = 2.35, and fc = 3.2. With the forcing function
163
¶
i
2
3
F (z, t) =
z (1 − z) + (3z − 2z − 1) e−t
6
¡
¢
fc
−
(1 + 3te1−t ) (2 − 12z + 12z 2 ) + i(1 − 2z) (1 − e−t )
2
2κ
¶
µ
i
2
3
2
2
− ik2 fc z (1 − z) + (3z − 2z − 1) (1 − e−t ),
6
µ
2
2
the complex-valued solution is
w(z, t) =
µ
¶
i
2
3
z (1 − z) + (3z − 2z − 1) (1 − e−t ).
6
2
2
This example illustrates a solution whose real and imaginary parts behave in a significantly different manner. In additional, the steady-state is nonzero. The discrete
system given by (4.130) is solved for the numerical solution. We use the parameter
choices α = β = γ = 1 in (4.90). The error results are reported in Table 28, Table
29. The graphical results for real and imaginary parts of the approximate solution
are shown in Figure 76, Figure 77, Figure 78, and Figure 79, respectively. Table
28 again suggests that we choose Nt = (1/2)Mz . Table 29 shows the errors with
Mz = Nz = Mt = 2Nt for Mz = 4, 8, 16, 32. The time plots for real and imaginary
parts of the approximate solution are graphed for z = 0, .16, .50 and 1 in Figure 80
and Figure 81 by using the values on the sinc grid S t . This time plots for real and
imaginary parts of the approximate solution are graphed for z = 0, .20, .50 and 1 in
Figure 82 and Figure 83 by using the values on the uniform grid U t .
164
Mz
Nz
Mt
Nt
h
kUS k
kVS k
kES k
kEU k
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
4
4
4
4
8
8
8
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
1
2
3
4
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1.111
1.111
1.111
1.111
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.785
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.555
0.003
0.003
0.003
0.003
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
3.090e-03
3.102e-03
3.100e-03
3.099e-03
6.443e-04
7.248e-04
7.119e-04
7.152e-04
7.146e-04
7.146e-04
7.146e-04
7.145e-04
2.562e-04
1.279e-04
6.302e-05
7.159e-05
6.658e-05
6.889e-05
6.818e-05
6.850e-05
6.839e-05
6.844e-05
6.842e-05
6.843e-05
6.843e-05
6.843e-05
6.843e-05
6.843e-05
3.090e-03
3.102e-03
3.100e-03
3.099e-03
6.443e-04
7.248e-04
7.119e-04
7.152e-04
7.146e-04
7.146e-04
7.146e-04
7.145e-04
4.593e-04
3.491e-04
1.433e-04
7.159e-05
6.658e-05
6.889e-05
6.818e-05
6.850e-05
6.839e-05
6.844e-05
6.842e-05
6.843e-05
6.843e-05
6.843e-05
6.843e-05
6.843e-05
3.099e-03
3.795e-03
3.614e-03
3.660e-03
1.409e-02
1.175e-03
8.494e-04
7.194e-04
7.518e-04
7.365e-04
7.424e-04
7.400e-04
2.464e-02
1.278e-02
8.087e-04
2.028e-04
1.189e-04
8.773e-05
7.401e-05
7.304e-05
6.835e-05
6.995e-05
6.884e-05
6.942e-05
6.912e-05
6.928e-05
6.919e-05
6.924e-05
Table 28. Errors on the sinc grid S and the uniform grid U for Mz = Nz = Mt and
various choices of Nt for Example 19 with Av (t) = 1 + te1−t and σ = 0.1.
165
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
h
kUS k
kVS k
1.111 0.003 3.102e-03
0.785 0.001 7.152e-04
0.555 0.000 6.850e-05
0.393 0.000 2.070e-06
kES k
kEU k
3.102e-03 3.795e-03
7.152e-04 7.194e-04
6.850e-05 7.304e-05
2.070e-06 2.214e-06
Table 29. Errors on the sinc grid S and the uniform grid U for the choices Mz =
Nz = Mt = 2Nt for Example 19 with Av (t) = 1 + te1−t and σ = 0.1.
PSfrag replacements
0.07
0.06
ua (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 76. The graph of the approximate solution (real part) ua (z, t) on the sinc grid
S for Example 19 with Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
166
0
va (z, t)
-0.05
-0.1
-0.15
-0.2
1
0.8
10
0.6
z
8
6
0.4
4
0.2
t
2
0
PSfrag replacements
0
Figure 77. The graph of the approximate solution (imaginary part) va (z, t) on the sinc
grid S for Example 19 where Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8.
0.07
0.06
ua (z, t)
0.05
0.04
0.03
0.02
0.01
0
1
0.8
10
0.6
z
8
6
0.4
4
0.2
2
0
t
0
Figure 78. The graph of the approximate solution (real part) ua (z, t) on the uniform
grid U for Example 19 with Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt = 8.
PSfrag replacements
167
0
va (z, t)
-0.05
PSfrag replacements
-0.1
1 -0.15
-0.2
1
0.8
10
0.6
z
8
6
0.4
4
0.2
t
2
0
0
Figure 79. The graph of the approximates solution (imaginary part) va (z, t) on the
uniform grid0.2U for Example 19 where Av (t) = 1 + te1−t with Mz = Nz = Mt =
16, Nt = 8. 0.4
0.6
0.8
0.07
0.06
ua (z, t)
0.05
z
z
z
z
0.04
=0
= .16
= .50
=1
0.03
0.02
0.01
0
0
1
2
3
4
5
6
7
8
9
10
t
Figure 80. The approximate solution (real part) ua (z, t) at z = 0, .16, .50, 1 on the
sinc grid S t for Example 19 where Av (t) = 1 + 3te1−t with Mz = Nz = Mt = 16, Nt =
8.
168
0
z
z
z
z
-0.02
=0
= .16
= .50
=1
-0.04
-0.06
va (z, t)
-0.08
PSfrag replacements
-0.1
-0.12
1
-0.14
-0.16
-0.18
-0.2
1
0
2
4
3
5
6
8
7
9
10
t
Figure 81. The approximate solution (imaginary part) va (z, t) at z = 0, .16, .50, 1
on the sinc grid S t for Example 19 where Av (t) = 1 + te1−t with Mz = Nz = Mt =
16, Nt = 8. 0.2
0.4
0.6
0.07
0.8
0.06
ua (z, t)
0.05
0.04
0.03
0.02
z
z
z
z
0.01
0
0
1
2
3
4
5
6
7
=0
= .20
= .50
=1
8
9
10
t
Figure 82. The approximate solution (real part) ua (z, t) at z = 0, .20, .50, 1 on the
uniform grid U t for Example 19 where Av (t) = 1 + 3te1−t with Mz = Nz = Mt =
16, Nt = 8.
169
0
z
z
z
z
-0.02
=0
= .20
= .50
=1
-0.04
-0.06
va (z, t)
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
0
1
2
3
4
5
6
7
8
9
10
t
Figure 83. The approximate solution (imaginary part) va (z, t) at z = 0, .20, .50, 1
on the uniform grid U t for Example 19 where Av (t) = 1 + 3te1−t with Mz = Nz =
Mt = 16, Nt = 8.
170
CHAPTER 5
THE SPIN-UP AND EPISODIC WIND STRESS PROBLEMS
In this chapter, we want to determine the performance of the fully Sinc-Galerkin
method, described in Chapter 4, when applied to realistic oceanography problems.
The complex velocity formulation for a time-dependent wind-driven problem is used.
A fully Sinc-Galerkin scheme is applied. To describe a realistic oceanography problem,
the eddy viscosity is assumed to depend on wind stress that is a function of time.
Let w(0) be a starting approximate solution for a small value of Mz . Then approximate solutions w (n) , w(n+1) are computed by the fully Sinc-Galerkin method in
(4.130), where Mz for w(n+1) is twice as large as that w (n) . Define kCU k, kCV k, and
kCM k as
©
ª
kCU k = max |R(w(n) (zm , tn )) − R(w (n+1) (zm , tn ))| ,
U
©
ª
kCV k = max |I(w(n) (zm , tn )) − I(w (n+1) (zm , tn ))| ,
U
kCM k = max {kCU k, kCV k} .
(5.1)
where the uniform grid U is defined in (4.92) with step size lz = 0.01, lt = 0.1. Graphs
are shown on the grids U and U t in (4.94) and (4.96), respectively. Remember
that R(w(z, t)) = u(z, t), the transformed (4.26) nondimensional northward current
velocity component, and I(w(z, t)) = v(z, t), the transformed (4.26) nondimensional
eastward current velocity component.
171
At a latitude of 50◦ N , a 60 meter deep coastal sea has a residual turbulence
corresponding to a background turbulent exchange coefficient A0 ≡ 0.02 m2 /s. This
is chosen as the scaling factor for the dimensional eddy viscosity. A storm with wind
directed at angle χ(t∗ ) measured clockwise from the north starts at t∗ = 0. The wind
stress varies as ψ(t∗ ) with the fundamental time scale of the forcing wind given by
T0 .
Suppose that wind-mixing throughout the water depth is reflected in an enhancement of the eddy viscosity, A∗v (z ∗ , t∗ ). Field evidence suggests that the near-surface
value of eddy viscosity is related to the wind stress, which depends on time. When the
sea is fairly shallow (D0 < 100 m), the eddy viscosity is assumed to be independent
of depth and to depend mainly on wind-mixing throughout the water column as
A∗v (t∗ ) = A0 [1 + kw ψ(t∗ )] ,
where kw is a magnification parameter and ψ(t∗ ) is normalized. In this case we choose
kw = 3.
The governing equations and variables in the subsequent development were nondimensionalized. The operative constants in (4.34)-(4.37) are κ, σ, and fc . The parameters and constants of nature in Chapter 3 are used. The fully Sinc-Galerkin method,
which is developed in Chapter 4, is used to find the numerical solution. Thus approximated are the time evolution of circulation features, such as velocity components,
speed, and the current profile. All numerical simulations were run on a SUN BLADE
172
1000 with MATLAB version 6.1. Throughout, comparable graphs are shown on the
same scale.
These simulations of realistic oceanography problems are carried out using parameters similar to those that have been used in earlier studies. The performance
of the method is determined by reporting the quantities kCU k, kCV k, and kCM k in
(5.1). Also, the numerical results were transformed by (4.26) and then multiplied by
the natural velocity scale U0 to give a dimensional representation of the velocities for
graphical representations.
The Spin-Up Wind Stress Problem
The fully Sinc-Galerkin scheme for the time-dependent eddy viscosity can determine the behavior of a realistic oceanography problem describing a storm where a
spin-up wind stress is given by
ψ(t∗ ) = 1 − e
−
³ ∗
t /T0
´
,
which is shown in Figure 84. T0 is shosen to be the time for the storm to build to
67% of its peak intensity.
Example 20. (Spin-up wind stress problem) Consider a nondimensional time-dependent
eddy viscosity Av (t) = 4 − 3e−t and a nondimensional wind stress ψ(t) = 1 − e−t . We
choose χ(t∗ ) ≡ π/4, a no-slip bottom condition, σ = 0, and D0 = 60 m. Parameter
choices are κ = 3.14, k2 = 1, and fc = T0 f = 4.95, where the fundamental time scale
10
173
1
0.9
Spin-up wind ψ(t∗ )
0.8
0.7
0.6
0.5
-0.05
0.4
-0.1
0.3
-0.15
0.2
-0.2
0.1
0
0
1
0.5
T0 = .45
1.5
2
2.5
3
3.5
4
4.5
x 105
Time t∗ (s)
∗
−
³ ∗
t /T0
Figure 84. The graph of the spin-up wind stress ψ(t ) = 1 − e
12.5 × 3600 = .45 × 105 s is the fundamental time scale of 12.5 hrs..
´
where T0 =
of the forcing wind is chosen to be 12.5 hours so T0 = 12.5 × 3600 = 4.5 × 104 s. The
initial-boundary-value problem (4.34)-(4.37) becomes
µ
¶
∂w(z, t)
fc ∂
−t ∂w(z, t)
− 2
(4 − 3e )
− ik2 fc w(z, t) = −κ(1 − z)r(t)
∂t
2κ ∂z
∂z
where r(t) is in (4.33). The surface and seabed boundary conditions and initial
condition are
∂w(0, t)
= 0, 0 < t,
∂z
w(1, t) = 0, 0 < t,
w(z, 0) = 0, 0 < z < 1.
Figure 85 shows the nondimensional spin-up wind stress. Here t0 corresponds to
the time for the storm to reach 67% of it peak intensity. For the fully Sinc-Galerkin
174
1
1
0.9
0.8
0.7
ψ(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
t0 = 1
5
6
7
8
9
10
t
Figure 85. The graph of the nondimensional spin-up wind stress ψ(t) = 1 − e−t where
T0 = 12.5 hrs. is the fundamental time scale, so t0 = 1.
scheme, the values Mz = Nz = Mt = 2Nt correspond to a discrete system of size
(mz mt ) × (mz mt ),
(mz = Mz + Nz + 3,
mt = Mt + Nt + 2) given in (4.130).
The convergence results in (5.1) are reported in Table 30 and shown in Figure 86,
Figure 87, Figure 88, and Figure 89, respectively. The values z ∗ = 0, 19, 30, and
60 m in Figure 88 and Figure 89 correspond to z ∗ = 0 (sea surface), DE (Ekman
depth), D0 /2 (middle ocean depth), and D0 (seabed), respectively. The approximate
northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc grid S is graphed in
Figure 86 using the values on the uniform grid U . The approximate eastward current
component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is graphed in Figure 87 using the
values on the uniform grid U .
175
h
kCU k
kCV k
kCM k
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2 1.111
4 0.785 1.293e-02 9.112e-03 1.293e-02
8 0.555 6.216e-03 4.770e-03 6.216e-03
16 0.393 3.181e-03 3.209e-03 3.209e-03
Table 30. The convergence results on the uniform grid U for the choices Mz = Nz =
Mt = 2Nt for Example 20 with Av (t) = 4 − 3e−t and σ = 0, κ = 3.14, fc = 4.95, and
D0 = 60 m.
PSfrag replacements
0
-0.005
Ua∗ (z ∗ , t∗ )
-0.01
-0.015
-0.02
-0.025
-0.03
60
50
5
40
4
30
Depth z ∗ (m)
3
20
2
10
1
0
5
x 10
Time
t∗
(s)
Figure 86. The graph of the approximate northward current component Ua∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 20 where Av (t) = 4 − 3e−t with Mz = Nz =
Mt = 16, Nt = 8.
176
0.08
0.07
Va∗ (z ∗ , t∗ )
0.06
0.05
0.04
0.03
0.02
0.01
0
60
50
×
5
40
4
30
3
20
Depth z ∗ (m)
2
10
5
x 10
1
0
Time
t∗
(s)
Figure 87. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 20 where Av (t) = 4 − 3e−t with Mz = Nz =
Mt = 16, Nt = 8.
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 88 using the values on the uniform grid U t . The time
plot is graphed for z ∗ = 0, 19, 30, and 60 m which correspond to 0 (sea surface),
DE (Ekman depth), (1/2)D0 (middle sea depth), and D0 (seabed), respectively. The
approximate eastward current component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is
graphed in Figure 89 using the values on the uniform grid U t . The time plot is
graphed for z ∗ = 0, 19, 30, and 60 m.
177
10
0
z∗
z∗
z∗
z∗
×
-0.005
=
=
=
=
0
19
30
60
Ua∗ (z ∗ , t∗ )
-0.01
PSfrag replacements
-0.015
-0.02
-0.025
-0.03
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 105
Time t∗ (s)
Figure 88. The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 20 where Av (t) = 4 − 3e−t with
Mz = N z = M
× t = 16, Nt = 8.
10
0.08
0.07
0.06
Va∗ (z ∗ , t∗ )
0.05
0.04
0.03
z∗
z∗
z∗
z∗
0.02
=
=
=
=
0
19
30
60
0.01
0
0
0.5
1
1.5
2
Time
2.5
t∗
(s)
3
3.5
4
4.5
x 105
Figure 89. The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 20 where Av (t) = 4 − 3e−t with
Mz = Nz = Mt = 16, Nt = 8.
178
Numerical Steady-State Problem for a No-Slip Bottom Condition
For a no-slip bottom condition (σ = 0), we can determine the accuracy of the fully
Sinc-Galerkin method at steady state (t → ∞) by comparing it to the true solution of
the steady-state problem. For Example 20, the fully Sinc-Galerkin discrete system of
size (mz mt )×(mz mt ), (mz = Mz +Nz +3, mt = Mt +Nt +2) given in (4.130) is first
used to compute the numerical solution generated on the sinc grid S. For the steadystate problem, the nondimensional time-dependent eddy viscosity Av (t) = 4 − 3e−t
becomes A∞ ≡ 4. So the steady-state boundary-value problem is
d2 w(z)
A∞
+ 2iκ2 w(z) = −2iκ3
dz 2
µ
1−z
A∞
¶
eiχ ,
(5.2)
with surface and seabed conditions
dw(0)
= 0,
dz
(5.3)
w(1) = 0.
(5.4)
With W (z) = w(z)+κ(1−z)eiχ , the true steady-state solution is W (z) = U0 (U (z) + iV (z))
where U (z) and V (z) are given by
U (z) = R(Wc (z))cos(χ) − I(Wc (z))sin(χ),
V (z) = R(Wc (z))sin(χ) + I(Wc (z))cos(χ).
The notations R(Wc (z)) and I(Wc (z)) denote the real part and imaginary part of
Wc (z), respectively, and
Wc (z) =
µ
³
q ´
1
¶ sinh (1 − i)κ(1 − z)
A∞
1+i
³
q ´.
√
2
A∞ cosh (1 − i)κ A1∞
(5.5)
179
This is very similar to Example 1. The numerical solution which is generated by the
sinc grid S is evaluated on the uniform grid U by increasing Mz = Nz = Mt = 2Nt =
4, 8, 16, 32 and choosing one large fixed time tT . In this case, we choose tT = 110
as a numerical representation of t → ∞. So the fully Sinc-Galerkin approximate
Wa (zj , tT ) = Ua (zj , tT ) + iVa (zj , tT ) is compared with the true steady-state solution
for U (zj ) and V (zj ) at the uniform grid Uz in (5.7). These results were then multiplied
by the natural velocity scale U0 to give a dimensional representation of the velocities.
To illustrate the performance of the method, the maximum absolute errors are defined
by
kUUz k = max {|Ua (zm , tT ) − U (zm )|} ,
Uz
kVUz k = max {|Va (zm , tT ) − V (zm )|} ,
Uz
kEUz k = max {kUUz k, kVUz k} ,
(5.6)
where the uniform grid Uz with step size lz = 0.01 is given by
Uz = {zm = mlz , m = 0, . . . , 100}.
(5.7)
Graphs are plotted on the uniform grid
Uz0 = {zm = mlz , m = 0, . . . , 40 and lz0 = 0.025}.
The errors are given in Table 31 and show a considerable degree of accuracy.
(5.8)
Figure
90 shows the numerical convergence of the fully Sinc-Galerkin method to the true
steady-state solution with the no-slip bottom condition (σ = 0), as Nz is repeatedly
180
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
tT
h
kUUz k
kVUz k
kEUz k
110 1.111 1.571e-02 5.340e-03 1.571e-02
110 0.785 5.498e-03 2.037e-03 5.498e-03
110 0.555 4.699e-04 1.089e-05 4.699e-04
110 0.393 1.016e-05 3.820e-05 3.820e-05
PSfrag replacements
Table 31. The errors of Example 20 (at steady-state) on the uniform grid Uz for the
choices Mz = Nz = Mt = 2Nt = 4, 8, 16, 32 and tT = 110 with a no-slip bottom
condition (σ = 0), κ = 3.14, fc = 4.95, and D0 = 60 m.
-3
x 10
N =4
N =8
N = 16
N = 32
True
5
Northward current component U ∗ (m/s)
10
×
0
z∗ = 0
z ∗ = 60
-5
-10
-15
-20
-25
-0.01
0
0.01
0.02
0.03
Eastward current component V
∗
0.04
0.05
(m/s)
Figure 90. Fully Sinc-Galerkin Ekman spiral projections for Example 20 on the
uniform grid Uz0 with increasing N ≡ Mz = Nz = Mt = 2Nt for the steady-state with
no-slip bottom boundary condition σ = 0 with χ = π/4, κ = 3.14, D0 = 60 m, DE =
19 m.
181
doubled in size. The horizontal projection of the Ekman spiral projection for N z = 32
is not indistinguishable from the true solution as shown by the solid line.
Example 21. (Spin-up wind stress problem) In Example 20 we replace the no-slip
bottom condition with the linear stress condition at the seabed, with σ = 0.16,
κ = 3.14, k2 = 1, and fc = T0 f =4.95, where the fundamental time scale of the
forcing wind is chosen to be 12.5 hours so T0 = 12.5 × 3600 = 4.5 × 104 s. The
initial-boundary-value problem (4.34)-(4.37) becomes
µ
¶
fc ∂
∂w(z, t)
−t ∂w(z, t)
(4 − 3e )
− ik2 fc w(z, t) = −κ(1 − z)r(t) − κσg(t)
− 2
∂t
2κ ∂z
∂z
where r(t) and g(t) are in (4.33). The surface and seabed boundary conditions and
initial condition are
∂w(0, t)
= 0, 0 < t,
∂z
∂w(1, t)
w(1, t) + σ(4 − 3e−t )
= 0, 0 < t,
∂z
w(z, 0) = 0, 0 < z < 1.
For the fully Sinc-Galerkin method, the values Mz = Nz = Mt = 2Nt correspond to
a discrete system of size (mz mt ) × (mz mt ), (mz = Mz + Nz + 3, mt = Mt + Nt + 2)
given in (4.130). The convergence results in (5.1) are reported in Table 32 and
shown in Figure 91, Figure 92, Figure 93, and Figure 94, respectively. The values
z ∗ = 0, 19, 30, and 60 m in Figure 93 and Figure 94 correspond to z ∗ = 0 (sea surface),
DE (Ekman depth), D0 /2 (middle ocean depth), and D0 (seabed), respectively.
182
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 91 using the values on the uniform grid U . The approximate eastward current component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is graphed
in Figure 92 using the values on the uniform grid U .
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 93 using the values on the uniform grid U t . The time
plot is graphed for z ∗ = 0, 19, 30, and 60 m which correspond to 0 (sea surface),
DE (Ekman depth), (1/2)D0 (middle sea depth), and D0 (seabed), respectively. The
approximate eastward current component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is
graphed in Figure 94 using the values on the uniform grid U t . The time plot is
graphed for z ∗ = 0, 19, 30, and 60 m.
h
kCU k
kCV k
kCM k
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2 1.111
4 0.785 1.177e-02 1.083e-02 1.177e-02
8 0.555 7.491e-03 6.711e-03 7.491e-03
16 0.393 6.541e-03 5.117e-03 6.541e-03
Table 32. The convergence results on the uniform grid U for the choices Mz = Nz =
Mt = 2Nt for Example 21 with Av (t) = 4 − 3e−t and σ = 0.16, κ = 3.14, fc = 4.95,
and D0 = 60 m.
183
×
0
-0.01
Ua∗ (z ∗ , t∗ )
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
60
50
5
40
4
30
PSfrag replacements
Depth z ∗ (m)
3
20
2
10
5
x 10
Time t∗ (s)
1
0
Figure 91. The graph of the approximate northward current component Ua∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 21 where Av (t) = 4 − 3e−t with Mz = Nz =
Mt = 16, Nt = 8.
×
0.14
0.12
Va∗ (z ∗ , t∗ )
0.1
0.08
0.06
0.04
0.02
0
60
50
5
40
4
30
Depth z ∗ (m)
3
20
2
10
1
0
5
x 10
Time t∗ (s)
Figure 92. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 21 where Av (t) = 4 − 3e−t with Mz = Nz =
Mt = 16, Nt = 8.
10
184
×
0
z∗
z∗
z∗
z∗
-0.01
=
=
=
=
0
19
30
60
Ua∗ (z ∗ , t∗ )
-0.02
-0.03
-0.04
-0.05
PSfrag replacements
-0.06
0
-0.07
-0.08
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 105
Time t∗ (s)
Figure 93. The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 21 where Av (t) = 4 − 3e−t with
Mz = Nz = Mt = 16, Nt = 8.
10
0.14
0.12
Va∗ (z ∗ , t∗ )
0.1
0.08
0.06
×
z∗
z∗
z∗
z∗
=
=
=
=
Time
t∗
0.04
0
19
30
60
0.02
0
0
0.5
1
1.5
25
2.5
(s)
3
3.5
4
4.5
x 105
Figure 94. The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 21 where Av (t) = 4 − 3e−t with
Mz = Nz = Mt = 16, Nt = 8.
185
Numerical Steady-State Problem for a Linear Stress Bottom Condition
For the linear stress condition at the seabed (σ = 0.16), the steady-state boundaryvalue-problem becomes
d2 w(z)
+ 2iκ2 w(z) = −2iκ3
A∞
2
dz
µ
¶
1−z
+ σ eiχ ,
A∞
(5.9)
with surface and seabed conditions
dw(0)
= 0,
dz
dw(1)
w(1) + σA∞
= 0.
dz
(5.10)
(5.11)
With W (z) = w(z)+κ(1−σ−z)eiχ , the true steady-state solution W (z) = U0 (U (z) + iV (z))
where U (z) and V (z) are given by
U (z) = R(Wc (z))cos(χ) − I(Wc (z))sin(χ),
V (z) = R(Wc (z))sin(χ) + I(Wc (z))cos(χ).
The notations R(Wc (z)) and I(Wc (z)) denote the real part and imaginary part of
Wc (z), respectively, and
Wc (z) =
κ (sinh(ω(1 − z)) + σA∞ ωcosh(ω(1 − z)))
A∞ ω (cosh(ω) + σA∞ ω sinh(ω))
where
ω=κ
r
−2i
.
A∞
(5.12)
186
Again this is very similar to Exaple 1. The fully Sinc-Galerkin discrete system
of size (mz mt ) × (mz mt ),
(mz = Mz + Nz + 3,
mt = Mt + Nt + 2) given in
(4.130) is used to compute the numerical solution generated by the sinc grid S.
After that, the numerical solution is evaluated on the uniform grid U by increasing
Mz = Nz = Mt = 2Nt = 4, 8, 16, 32 and choosing one large fixed time tT = 110. So
the fully Sinc-Galerkin approximate Wa (zj , tT ) = Ua (zj , tT ) + iVa (zj , tT ) is compared
with the true steady-state solution for U (zj ) and V (zj ) at the uniform grid Uz . These
results were then multiplied by the natural velocity scale U0 to give a dimensional
representation of the velocities. To illustrate the performance of the method, the
maximum absolute errors are reported. The errors are given in Table 33 and show an
accuracy similar to that of Example 20 in Table 31. Figure 90 shows the Sinc-Galerkin
Ekman spiral projection.
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2
4
8
16
tT
h
kUUz k
kVUz k
kEUz k
110 1.111 2.517e-02 1.495e-02 2.517e-02
110 0.785 9.327e-03 6.186e-03 9.327e-03
110 0.555 8.068e-04 5.053e-04 8.068e-04
110 0.393 5.639e-05 7.258e-05 7.258e-05
Table 33. The errors of Example 21 (at steady-state) on the uniform grid Uz for the
choices Mz = Nz = Mt = 2Nt = 4, 8, 16, 32 and tT = 110 with the linear stress
bottom condition (σ = 0.16), κ = 3.14, fc = 4.95, and D0 = 60 m.
187
-3
x 10
N =4
N =8
N = 16
N = 32
True
5
Northward current component U ∗ (m/s)
10
×
0
z∗ = 0
-5
z ∗ = 60
-10
-15
-20
-25
-0.01
0
0.01
0.02
0.03
0.04
0.05
Eastward current component V ∗ (m/s)
Figure 95. Fully Sinc-Galerkin Ekman spiral projections for Example 21 (for the
steady-state) on the uniform grid Uz0 with increasing N = Mz = Nz = Mt = 2Nt with
the linear stress bottom boundary condition (σ = 0.16), χ = π/4, κ = 3.14, D 0 =
60 m, DE = 19 m.
The Episodic Wind Stress Problem
In the second case, a storm with an episodic wind stress is given by
ψ(t∗ ) =
t∗ 1− Tt∗
e 0
T0
which is shown in Figure 96. The time scale T0 used is the time for the storm to reach
its peak intensity. The fully Sinc-Galerkin scheme is used to determine the numerical
approximation.
1
188
1
0.9
Episodic wind ψ(t∗ )
0.8
0.7
3.5
4
0.6
4.5
0.5
0.4
-0.05
-0.1
-0.15
0.3
0.2
-0.2
0.1
0
0
0.5
T0 = .29
1
1.5
2
Time t∗ (s)
2.5
3
x 105
∗
Figure 96. The graph of the episodic wind stress ψ(t∗ ) = (t∗ /T0 )e1−(t /T0 ) where
T0 = 8.1 × 3600 = 2.916 × 104 s is the fundamental time scale of 8.1 hours.
Example 22. (Episodic wind stress problem) Consider a nondimensional time-dependent
eddy viscosity Av (t) = 1 + 3te1−t and a nondimensional wind stress ψ(t) = te1−t . We
choose χ(t∗ ) ≡ π/4, a no-slip condition at the seabed, σ = 0, D0 = 60 m. Parameter
choices are κ = 3.14, k2 = 1, and fc = T0 f = 3.2, where T0 , the fundamental time
scale of the forcing wind, is chosen to be 8.1 hours as T0 = 8.1×3600 s (= 2.916×104 s).
The initial-boundary-value problem (4.34)-(4.37) becomes
µ
¶
∂w(z, t)
fc ∂
1−t ∂w(z, t)
− 2
(1 + 3te )
− ik2 fc w(z, t) = −κ(1 − z)r(t)
∂t
2κ ∂z
∂z
where r(t) is in (4.33). The surface and seabed boundary conditions and initial
condition are
189
PSfrag replacements
∂w(0, t)
= 0, 0 < t,
∂z
1
w(1, t) = 0, 0 < t,
w(z, 0) = 0, 0 < z < 1.
Figure 97 shows the nondimensional episodic wind stress. Here t0 corresponds to the
time for the storm to reach its peak intensity.
1
0.9
0.8
0.7
ψ(t)
0.6
0.5
0.4
-0.05
-0.1
-0.15
-0.2
0.3
0.2
0.1
0
0
1
t0 = 1
2
3
4
5
6
7
8
9
10
t
Figure 97. The graph of the nondimensional episodic wind stress ψ(t) = te1−t where
T0 = 8.1 hrs. is the fundamental time scale, so t0 = 1.
The values Mz = Nz = Mt = 2Nt of the Sinc-Galerkin scheme correspond to a
discrete system of size (mz mt ) × (mz mt ), (mz = Mz + Nz + 3, mt = Mt + Nt + 2)
given in (4.130). The convergence results in (5.1) are reported in Table 34 and shown
in Figure 98, Figure 99, Figure 100, and Figure 101, respectively.
190
h
kCU k
kCV k
kCM k
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2 1.111
4 0.785 3.812e-02 5.524e-02 5.524e-02
8 0.555 1.477e-02 1.580e-02 1.580e-02
16 0.393 5.975e-03 8.106e-03 8.106e-03
Table 34. The convergence results on the uniform grid U for the choices Mz = Nz =
Mt = 2Nt for Example 22 with Av (t) = 1 + 3te1−t and σ = 0, κ = 3.14, fc = 3.2, and
D0 = 60 m.
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 98 using the values on the uniform grid U .
The
approximate eastward current component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is
graphed in Figure 99 using the values on the uniform grid U .
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 100 using the values on the uniform grid U t . The time
plot is graphed for z ∗ = 0, 19, 30, and 60 m. The approximate eastward current
component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is graphed in Figure 101 using the
values on the uniform grid U t . The time plot is graphed for z ∗ = 0, 19, 30, and
60 m.
191
0
-0.005
-0.01
Ua∗ (z ∗ , t∗ )
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
PSfrag replacements
-0.045
60
50
3
40
2.5
30
0
Depth z ∗ (m)
2
1.5
20
1
10
0.5
0
5
x 10
Time t∗ (s)
Figure 98. The graph of the approximate northward current component Ua∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 22 where Av (t) = 1 + 3te1−t with Mz =
Nz = Mt = 16, Nt = 8.
0.08
0.07
Va∗ (z ∗ , t∗ )
0.06
0.05
0.04
0.03
0.02
0.01
0
60
50
3
40
2.5
30
Depth z ∗ (m)
2
1.5
20
1
10
0.5
0
Time t∗ (s)
5
x 10
Figure 99. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 22 where Av (t) = 1 + 3te1−t with Mz =
Nz = Mt = 16, Nt = 8.
5
192
10
0
-0.005
-0.01
Ua∗ (z ∗ , t∗ )
-0.015
-0.02
-0.025
-0.03
z∗
z∗
z∗
z∗
-0.035
=
=
=
=
0
19
30
60
-0.04
PSfrag replacements
-0.045
0
0.5
1
2
1.5
2.5
3
x 105
Time t∗ (s)
0
Figure 100. The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 22 where Av (t) = 1 + 3te1−t
with Mz = Nz = Mt = 16, Nt = 8.
0.08
0.07
z∗
z∗
z∗
z∗
0.06
=
=
=
=
0
19
30
60
Va∗ (z ∗ , t∗ )
0.05
0.04
0.03
0.02
0.01
0
0
0.5
1
2
1.5
2.5
3
5
Time
t∗
(s)
x 10
Figure 101. The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 22 where Av (t) = 1 + 3te1−t
with Mz = Nz = Mt = 16, Nt = 8.
193
Example 23. ( Episodic wind stress problem) In Example 22 we replace the no-slip
bottom condition with the linear stress condition at the seabed (D0 = 60 m) with
σ = 0.16, κ = 3.14, k2 = 1, and fc = T0 f = 3.2, where T0 , the fundamental time scale
of the forcing wind, is chosen to be 8.1 hours so T0 = 8.1 × 3600 = 2.916 × 104 s. The
initial-boundary-value problem (4.34)-(4.37) becomes
µ
¶
∂w(z, t)
fc ∂
1−t ∂w(z, t)
− 2
(1 + 3te )
− ik2 fc w(z, t) = −κ(1 − z)r(t) − κσg(t)
∂t
2κ ∂z
∂z
where r(t) and g(t) are in (4.33). The surface and seabed boundary conditions and
initial condition are
∂w(0, t)
= 0, 0 < t,
∂z
∂w(1, t)
= 0, 0 < t,
w(1, t) + σ(1 + 3te1−t )
∂z
w(z, 0) = 0, 0 < z < 1.
For the computation, the values Mz = Nz = Mt = 2Nt of the Sinc-Galerkin scheme
correspond to a discrete system of size (mz mt ) × (mz mt ), (mz = Mz + Nz + 3, mt =
Mt + Nt + 2) given in (4.130). The convergence results in (5.1) are reported in Table
35 and shown in Figure 102, Figure 103, Figure 104, and Figure 105, respectively.
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 102 using the values on the uniform grid U .
The
approximate eastward current component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is
graphed in Figure 103 using the values on the uniform grid U .
194
h
kCU k
kCV k
kCM k
Mz
Nz
Mt
Nt
4
8
16
32
4
8
16
32
4
8
16
32
2 1.111
4 0.785 5.288e-02 8.268e-02 8.268e-02
8 0.555 3.028e-02 2.606e-02 3.028e-02
16 0.393 2.131e-02 2.606e-02 2.606e-02
Table 35. The convergence results on the uniform grid U for the choices Mz = Nz =
Mt = 2Nt for Example 23 with Av (t) = 1 + 3te1−t and σ = 0.16, κ = 3.14, fc = 3.2,
and D0 = 60 m.
PSfrag replacements
0.02
Ua∗ (z ∗ , t∗ )
0
-0.02
−0.04
-0.06
-0.08
-0.1
60
50
3
40
2.5
30
Depth z ∗ (m)
2
1.5
20
1
10
0.5
0
5
x 10
Time t∗ (s)
Figure 102. The graph of the approximate northward current component Ua∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 23 where Av (t) = 1 + 3te1−t with Mz =
Nz = Mt = 16, Nt = 8.
195
0.14
0.12
Va∗ (z ∗ , t∗ )
0.1
0.08
0.06
0.04
0.02
0
-0.02
60
50
3
40
Depth z ∗ (m)
2.5
30
2
1.5
20
1
10
0.5
0
5
x 10
Time t∗ (s)
Figure 103. The graph of the approximate eastward current component Va∗ (z ∗ , t∗ )
(m/s) on the uniform grid U for Example 23 where Av (t) = 1 + 3te1−t with Mz =
Nz = Mt = 16, Nt = 8.
The approximate northward current component Ua∗ (z ∗ , t∗ ) generated by the sinc
grid S is graphed in Figure 104 using the values on the uniform grid U t . The time
plot is graphed for z ∗ = 0, 19, 30, and 60 m. The approximate eastward current
component Va∗ (z ∗ , t∗ ) generated by the sinc grid S is graphed in Figure 105 using the
values on the uniform grid U t . The time plot is graphed for z ∗ = 0, 19, 30, and 60
m.
196
0.01
0
-0.01
Ua∗ (z ∗ , t∗ )
-0.02
-0.03
-0.04
-0.05
z∗
z∗
z∗
z∗
-0.06
=
=
=
=
0
19
30
60
-0.07
-0.08
PSfrag replacements
-0.09
0
0.5
1
2
1.5
2.5
3
5
x 10
Time t∗ (s)
Figure 104. The approximate northward current component Ua∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 23 where Av (t) = 1 + 3te1−t
with Mz = Nz = Mt = 16, Nt = 8.
0.14
0.12
z∗ = 0
z ∗ = 19
z ∗ = 30
z ∗ = 60
Va∗ (z ∗ , t∗ )
0.1
0.16
0.08
0.06
-0.18
0.04
0.02
0
-0.2
0
0.5
1
2
1.5
2.5
3
5
Time
t∗
(s)
x 10
Figure 105. The approximate eastward current component Va∗ (z ∗ , t∗ ) (m/s) at z ∗ =
0, 19, 30, 60 m on the uniform grid U t for Example 23 where Av (t) = 1 + 3te1−t
with Mz = Nz = Mt = 16, Nt = 8.
197
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