Sinc-Galerkin solution of second-order hyperbolic problems in multiple space dimensions
by Kelly Marie McArthur
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Mathematics
Montana State University
© Copyright by Kelly Marie McArthur (1987)
Abstract:
A fully Galerkin method in space and time is developed for the second-order hyperbolic problem in
one, two, and three space dimensions. Using sinc basis functions and the sinc quadrature rule, the
discrete system arising from the orthogonalization of the residual is easily assembled. (Two equivalent
matrix formulations of the systems are given. One lends itself to scalar computation while the other is
more natural in a vector computing environment. In fact, it is shown that passing from . one to the other
is simply a notational change. In either setting the move from one to two or three space dimensions
does not significantly affect the ease of implementation. Intermediate diagonalization of each matrix
representing the discretization of a second partial leads to the diagonalization of the overall system.
The method was tested on an extensive class of problems. Numerical results indicate the method has an
exponential convergence rate for analytic and singular problems. Moreover that is independent of the
spatial dimension. SINC-GALERKIN SOLUTION OF SECOND-ORDER HYPERBOLIC
PROBLEMS IN MULTIPLE SPACE DIMENSIONS
by
Kelly Marie McArthur
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Mathematics
MONTANA STATE UNIVERSITY
Bozeman, Montana
May 1987
£>37£
M ll
9f
C-Ojp,^
ii
APPROVAL
of a thesis submitted by
Kelly Marie McArthur
This thesis
has been
read by each member of the thesis
committee and
has been
found to be satisfactory regarding
c o n t e n t , English u s a g e , f o r m a t , citations, bibliographic
s t y l e , and consistency, and is ready
for submission
to the
College of Graduate Studies.
^-Q j m 7
Chairperson,
Graduate Committee
Date
Approved for the Major Department
JZd / 9 * 7
HeauTyMathema^icad
Sciences Department
Date
Approved for the College of Graduate Studies
jr-?A
Date
Graduate Dean
iii
STATEMENT OF PERMISSION TO USE
In presenting
requirements
University,
for
this thesis in partial fulfillment of the
a
doctoral
Montana
State
of the
Library.
I further agree
of this thesis is allowable only for scholarly
p u r p o s e s , consistent
U .S .
at
I agree that the Library shall make it available
to borrowers under rules
that copying
degree
Copyright
Law.
with "fair
Requests
use" as
for
prescribed in the
extensive copying or
reproduction of this thesis should be referred to University
Microfilms
I n ternational,
Michigan 48106,
to reproduce
300
North Zeeb Road, Ann Arbor,
to whom I have granted "the
and distribute
copies of
exclusive right
the dissertation in
and from microfilm and the right to reproduce and distribute
by abstract in any f o r m a t ."
Date
iv
ACKNOWLEDGMENTS
The author thankfully acknowledges the superb typing of
Ms. R e n e 1
Tritz and
Eric G r e e n w a d e .
for
his
the invaluable
Thanks also go
careful
reading
of
to Professor
this w o r k .
author recognizes Professor Frank
and extension
of sine
Stenger
function t h e o r y .
and willingness to discuss
another generation
computer assistance of
new
problems
Norman Eggert
In addition,
for
the
the revival
His clear exposes
have
of sine function s t u d e n t s .
led
to yet
Finally,
the
author thanks Professor Kenneth L . Bowers and Professor John
Lund,
whose
guidance
is
responsible for this work.
possess the rare ability to teach one to teach oneself.
They
V
TABLE OF CONTENTS
Page
1.
2.
3.
4.
5.
I N T R O D U C T I O N.... ...................................
THE SINC-GALERKIN SOLUTION OF ORDINARY
DIFFERENTIAL EQUATIONS .........
I
8
Interpolation on (-=, = ) .........................
Quadrature on (-»,<»)............................
Interpolation and Quadrature
on Alternative A r c s . . . . . .................
Sinc-Galerkin M e t h o d ............................
9
18
22
26
THE SINC-GALERKIN METHOD
FOR THE WAVE E Q U A T I O N ..............................
35
SOLUTION OF THE DISCRETE SINC-GALERKIN SYSTEM...
56
Classical M e t h o d s ................................
Solution of the Discrete System in
One Spatial V a r i a b l e ................
Solution of the Systems in Two and
Three Spatial V a r i a b l e s .......................
. 56
61
65
NUMERICAL EXAMPLES O F THE SINC-GALERKIN M E T H O D . .
71
Exponentially Damped Sine W a v e s . ...............
Singular P r o b l e m s ..........
Problems Dictating Noncentered Sums
in All I n d i c e s . .........
78
82
REFERENCES CITED
86
91
vi
LIST OF TABLES
Table
1.
Page
Machine Storage Information for the
Matrices B (^ ) and B^J ) ...........................
58
2.
Numerical
Results for Example 5 . 1 ............
79
3.
Numerical
Results for Example 5 . 2 ...............
80
4.
Numerical
Results for Example 5 . 3 ...............
81
5.
Numerical Results for the Damped Sine W a v e ....
81
6.
Numerical
Results for Example 5 . 4 .......
82
7.
Numerical
Results for Example 5 . 5 ...............
84
Numerical Results for Example 5 . 6 ...............
85
8. .
/
9.
Numerical Results for the Singular Problems....
85
10.
Numerical
Results for Example 5 . 7 . . . . ...........
86
11.
Numerical
Results for Example 5 . 8 ...............
88
12.
Numerical
Results for Example 5 . 9 ...............
89
13.
Numerical Results for Problems Dictating
Nbncentered Sums in All Indices. . ...............
89
vii
LIST OF FIGURES
Figure
Page
1.
The Domain of D e p e n d e n c e ..........................
3
2.
The Numerical Domain of Dependence
for the E x p l i c i t , Centered Finite
Difference M e t h o d ..................................
4
3.
S (0,1 ) (x) = sine (x) , x 6 R ........ ..............
10
4.
The Domain D g ............... ..... . .................
14
5.
The Conformal Map X ........... ............... .
6.
The Regions Dg , Dw , and Dg ........................
38
7.
The Basis Functions S q (X) and S q (t ) ..............
39
.
23
viii
ABSTRACT
A fully
Galerkin method
in space and time is developed
for the second-order hyperbolic
problem
in one,
two, and
three space
dimensions.
Using sine basis functions and the
sine quadrature rule, the
discrete system arising from the
orthogonalization of
the residual is easily a s s e m b l e d . ^Two
equivalent matrix formulations
of
the
systems
are given.
One lends
itself to
scalar computation while the other is
more natural in a vector computing environment.
In fact, it
is shown
that
passing
from . one to the other is simply a
notational c h a n g e . In either setting
the move
from one to
two or
three space dimensions does not significantly affect
the ease of implementation.
Intermediate diagonalization of
each matrix
representing
the
discretization
of a second
partial leads to the diagonalization of the overall system.
The method was tested on an extensive class of p r o b l e m s .
Numerical
results
indicate
the
method has an exponential
convergence
rate
for analytic
and singular
problems.
Moreover that is independent of the spatial dimension.
CHAPTER I
INTRODUCTION
The
general
seco n d - o r d e r ,
linear partial differential
equation
(1.1)
Aux?c + Bux y + Cuyy + Dux + Euy + Fu = G
,
where A, B, C, D, E , F, and G are functions of x and y only,
is classified
at the point
(B2 - 4A C ) (x,y).
zero,
or
That is, when
negative
parabolic,
the
or elliptic,
reviews classical
(x,y) via the discriminate
(B2 - 4AC)(x,y)
equation
at
(x,y)
respectively.
The
is positive,
is hyperbolic,
present chapter
results for a purely hyperbolic,
constant
coefficient problem corresponding to A = -I, C = I, and
B
=
D
=
E
conditions,
= F
the
= 0 .
Including boundary and initial
specific
partial
differential
equation
examined is
u tt (x,t ) - u x x (x,t ) = G (x,t )
u(0,t) = u (I ,t ) = 0
(x,t ) E (0,1) x (0,“ )
t > 0
( 1 .2 )
u(x,0)
= f(x)
0 < x < I
Ut (XzO) = g(x)
This
problem
is
referred
equation since it models
string with
the
0 < x < I
to
as
a
one-dimensional wave
displacement
of
a vibrating
initial displacement f, initial velocity g, and
subject to an external force G .
2
The equation
in (1.2)
is representative
of one of the
two hyperbolic canonical forms due to the absence of the
term Ux t -
In
alternative
c o n t r a s t , the
canonical
form
distinguishing feature
is
term is the mixed partial U x t .
of the
that its only second-order
The change of variables
f = x + t and 'n = x - t transforms the equation in (1.2) to
(1.3)
-4u?)? = G((f + i?)/2 , (f -
Ti)/
2)
(?,%) G (0,=) x
It
is
well-known
that
found by solving two
depend on
and
t,
the correct change of variables is
ordinary differential
the discriminate.
Farlow [I].
and
By integrating
applying
(-<»,I)
equations which
For a complete discussion see
(1.3), restoring the variables x
the initial and boundary conditions,
d'Alembert explicitly solved (1.2)
[2].
His result
of 1747
is
u(x,t )
1
f (x + t) + f(x — t) +
x+t
J g (<X)da
2
x-t
(1.4)
t
where f ,
g, and
J
x + (t — p )
J
G(a,p)dadp
O
x-(t-p)
G are
extended as
odd periodic functions
when necessary.
The
solution
particular
where
(1.4)
solutions,
is
the
uH (x,t)
sum
and
of
homogeneous
uP (x,t),
and
respectively.
3
(1.5)
1
UH (X ,t )
2
x+t
f(x+t)
+f(x-t)
+
J
g (a )da
and
t
x+(t-p)
J
up (x ,t )
( 1 .6 )
G (a ,p )dadp
x-(t-p)
From the
form of
the homogeneous
that u H at a fixed point
As
a
result
interval of d e p e n d e n c e .
to find
[Xq
[xQ - t0 , X q + tQ ] on the
-
tQ , xQ + tQ ] is called the
The particular solution may be used
the domain of dependence shown by the shaded region
in Figure I below.
The
curves representing
f
and
n
that
the
characteristics
variables
Notice
it is apparent
(Xq ,t 0 ) depends only on the initial
conditions over the interval
x-axis.
solution,
dependence
out
of
are
the
called
x-axis
the change of
characteristics of
cut
and
the
define
(1.2).
interval
of
the domain of
dependence.
xn + t
Figure
The Domain of D e p e n d e n c e .
The significance of
constraint it
imposes on
the
domain
of
dependence
is the
schemes used to numerically solve
the one-dimensional wave equation
(1.2).
This constraint is
4
called the
A
scheme
Courant-Friedrichs-Lewy
which
condition is
provides
a
the centered
(or
ready
C F L ) condition [3].
illustration
of
the
finite difference method defined
explicitly by the difference equation
(1,7)
U i,j = m2(Ui+l,j-1
+ U i - l >j - l ) + 2(1 " m2)Ui,j-i
" U i,j-2 + (At)^ G (x i 't J - I ) •
Here U i ^ approximates u at
m = At/Ax
is
derivation of
Ames
[4].
difference
below).
the
ratio
(Xi ,tj) = (iAx,jat)
of
the
stepsizes.
the time-marching scheme
Stepping
back
gridpoints
which
These gridpoints
dependence outlined
in
time
influence
2.
(1.7)
A detailed
is included in
determines
blanket the
in Figure
and
U i j (see Figure 2
numerical domain of
The CFL criteria states
that a necessary condition for the convergence of
At and
must
Ax -»
contain
requires m
0) is
the
<1.
the finite
(1.7)
(as
that the numerical domain of dependence
analytic
domain
of
dependence.
This
The computational impact of restricting
m < I is briefly discussed in Chapter 4.
t. --
Figure 2.
The Numerical Domain of Dependence for the
E x p l i c i t , Centered Finite Difference Method.
5
The
method
condition
of
the
present
trivially.
Galerkin
method,
This
builds
Fourier
tensor products of
conformal maps.
(0,1) x
satisfies
technique,
called
the
The approximate
expansion
sine
CFL
the Sinc-
an approximate solution for
valid on the entire domain.
generalized
work
(1.2)
is a truncated
of basis elements which are
functions
composed
with suitable
,The support of each.basis element is
(0,») hence,
the method
may be
termed a spectral
method and its numerical domain of dependence is identically
the domain of the partial differential equation.
To
ease
applied to
the
description
(1.2),
equations is
of
the Sinc-Galerkin method
its counterpart for
derived in Chapter 2.
ordinary differential
Here the basis elements
are single sine functions composed with conformal maps.
pertinent
sine
function
properties needed to construct an
approximate solution are reviewed.
that when
2N +
approximate,
Ofe-K^fN"),
>
s i n g u lar i t i e s .
than that
I basis
Further,
functions are
o,
Stenger
occurs
even
it
is shown
used to define this
the optimal exponential order of
K
The
in
the
convergence,
presence
of
[5] discusses a more general setting
considered in Chapter 2.
The chapter closes with
the formulation of the discrete linear system whose solution
specifies
the
result that Lund
approximate.
system is symmetric,
a
[6] has shown depends on the correct choice
of weighted inner product.
i
)
This
6
Chapter
(with f
3
= g
equations
writing.
extends
two
and
most
the
Sinc-Galerkin
and then
= 0)
in
the
further to
three
common
space
method to (1.2)
the analogous wave
dimensions.
procedure
for
At this
solving partial
differential equations with a semi-infinite time interval is
a Galerkih
discretization of
the spatial
domain with time
dependent c o e f f i c i e n t s .
The result is a system of ordinary
differential
usually
equations
techniques.
Botha
discretizations
of
functions.
In
and
space
solved
Finder
[3]
using
contrast,
Gottlieb
and
consider only
However,
element
Orszag
basis
[7]
They show
use
that in
because
Gottlieb
and Orszag
finite difference techniques for the temporal
the time solution
acknowledge
Galerkin
of these spectral methods exhibit an exponential
convergence rate.
domain,
difference
develop
finite
globally defined spatial basis elements.
space many
via
the
has finite-order
incompatibility
accuracy.
They
of the error statement in
time versus space with the following remark:
No e f f i c i e n t , infinite-order accurate timedifferencing methods for variable coefficient
problems are yet k n o w n . The current
state-of-the-art of time-integration techniques
for spectral methods is far from satisfactory on
both theoretical and practical g r o u n d s . . .[7].
The point of view
just cited
time domain.
taken here
differs from
the two sources
by carrying the Galerkin discretization into the
Chapter
5
reports
numerical
attest to the success of this notion.
i
results which
7
Besides developing
the Sinc-Galerkin
introduces notation to
resulting discrete
facilitate
systems.
the
depends somewhat
Chapter 3
description
of the
These systems are posed in two
algebraically equivalent matrix f o r m s .
to use
method.
The
choice of form
on available computing facilities.
Chapter 4 discusses this topic along with algorithms for the
solution of the linear systems in either form.
The nine
groups of
examples included in Chapter 5 are broken into
three.
Sinc-Galerkin
method
space dimensions.
have
analytic
combinations
Each group
For
problems
instance,
solutions
of
singularities.
for
highlights a
both
in one,
the
while
feature of the
two, and three
first three examples
the
second
algebraic
and
three
logarithmic
The numerical results show that the rate of
convergence is not affected by this singular behavior.
last three
the
The
examples show the dramatic reduction in the size
of the discrete system
parameter
have
selections.
asymptotic
error
solved
when
Finally,
0(e"~^'^") ,
care
is
exercised in
each group indicates that
K
>
0,
is
independent of the dimension of the wave equation.
\
attained
8
CHAPTER 2
THE SINC-GALERKIN SOLUTION OF ORDINARY
DIFFERENTIAL EQUATIONS
The goal of this chapter is to derive the discrete SincGalerkin system necessary to
build
an
approximate
to the
solution of
Lf(x)
s f " (x) + v(x)f(x) = o(x)
a < x < b
(2 . 1 )
f (a) = f(b) = O
valid on the interval
(a ,b ).
A symmetric matrix formulation
for the system can be posed and i s ,
numerically.
The resulting
in f a c t ,
approximate solution converges
to the true solution on (a,b) at the
is a
positive constant
to build the approximate.
maintained
in
the
statements
analysis
a
of
where K
Further,
the convergence
rate is
presence of singularities' (the solution
theory is n e c e s s a r y .
error
rate 0 ( e _l<:JTr)
and 2N + I basis functions are used
has an unbounded derivative)
these
easy to solve
on
background
the
in
boundary.
general
sine function
In p a r t i c u l a r , the foundation
the
Sinc-Galerkin
To prove
for the
method is the error
associated w ith the truncated sine quadrature r u l e .
9
Interpolation on
Numerical
Whittaker's
at the
sine
function
methods
are
rooted
[8] work concerning interpolation of
integers.
Rather than
based on polynomials,
properties are
proper setting.
The
a function
using well-known expansions
Whittaker
far more
in E .T .
sought
an
distinguished when
foundation of
expansion whose
applied in the
the series
is the sine
function
sin(Tcx)
sine(x)
(2 .2 )
x E R
ItX
shown
in
Figure
3
below.
The resulting formal cardinal
series for a function f is
(2.3)
^
f(k)sine(x - k)
k=-«>
To generalize
(2.2) and
(2.3)
to handle interpolation on any
evenly spaced grid define for h > 0
sinc^-
S(k,h)(x)
(2.4)
and denote the Whittaker cardinal function by
(2.5)
C (f ,h ) (x)
E
Y
f (kh)S(k,h)(x)
k=-™
whenever this
(2.5)
series c o n v e r g e s .
In engineering literature
is often called a band-limited series.
10
Figure 3:
S ( 0 , I ) (x) = sine(x)
, x E R.
With regard to using (2.5) as an approximation t o o l , two
important classes of
functions
must
be
identified.
The
first is the very restricted class for which (2.5)
is exact.
The second
which the
is
the
difference between
class
f and C (f ,h)
[9] and M c N a m e e , et.al.
task
by
displaying
of
a
functions
is s m a l l .
[10] accomplish
natural
f
for
J.M. Whittaker
this identification
link between the Whittaker
series and aspects of Fourier series and i n t e g r a l s .
The Fourier transform for a function g is
( 2 .6 )
A fundamental result of Fourier analysis is that if
g E
L 2 (R) then
g E
L 2 (R) and g is recovered from g by the
Fourier inversion integral
ii
g( t)
(2.7)
For
a
select
= —
2ir
set
f
J
of
g(x)e ixtdx
functions,
the Paley-Wiener Theorem
shows that the inverse transform has compact s u p p o r t .
Their
result is
Theorem
(2.8):
If
g
G
L2 (R),
positive constants A and C so that
entire,
|g(w)|
and there exist
< C e x p { A | w | } where
w G (5> then
(2.9)
g(w)
I
A a
= 2 jt J g(x)e ixwdx
-A
.
Showing that the sine function satisfies the hypotheses of
Theorem
(2.8)
with A
= it and C = I is straightforward.
An
elementary calculation gives
(2.10)
sinc(w)
=
J
e " ixwdx
X (_ K z K )e ™ ixwdx ;
-it
hence,
- 00
an immediate consequence of Theorem (2.8)
is
CO
(2.11)
SincT(X)
= I
To accommodate the
(2.4),
(2.12)
sinc( t ) e ixtdt = X (-*,*) (x)
translated
sine
function
a change of variables in (2.10) gives
S (k ,h )(x )
J
sine
h
e ixtdt
appearing in
12
_ J1 e ixS
^ (-ir/h, K/h) (x)
for fixed real S .
The
support
of
the
characteristic function in (2.12)
prompts the definition of a class
Paley-Wiener c l a s s ,
which is
of functions,
called the
naturally associated with the
Whittaker series.
Definition
(2.13):
Let B (h) be the set of functions
that g 6 L2 (R), g is e n t i r e , and
|g(w)|
g such
< C e x p {tc |w|/h) where
w 6 0.
Before the Whittaker function
can be
discussed,
one result
is vital.
Theorem (2.14):
If g E B(h)
I
zt-w\
= - J g (t )sine
)d t .
then g(w)
'
-CD
A proof of Theorem (2.14)
is found in [10].
completeness the converse of Theorem
Theorem
(2.15):
(2.14)
If g E L2 (R) then k(w)
I
'
For
is
00
zt-w\
- r g ( t ) s i n e (-^-Idt
is in B ( h ) .
Proof:
Using P a r s e v a l 1s Theorem with the inner product
<f,g> = J
for k.
f (t ) g (t ) dt
establishes the growth estimate
Application of M o r e r a 1s Theorem proves
the Cauchy-Schwarz inequality yields k E L2 (R).
entirety and
13
The significance of all
the proceeding
work is
in the
then g(w) “
ak S(k,h)(w)
subsequent elegant theorem.
Theorem (2.16):
If g G B(h)
J
k=-®
where
Proof:
(2.17)
ak = h
J
t - kh
g(t)sinc^— ---- jdt = g(kh)
The Paley-Wiener Theorem,
e-ixw _ ^ sin
(?)Z
the identity
(-D
k
.ikhx
w+kh
k=-«
and the uniform convergence theorem justify the ensuing
steps:
g(w)
ir/h
J
g(x)e ixwdx
I
-rc/h
jr/h
sin
h
jr
h
re
Z
k=-®
L
k=-®
^
k=-®
(?)/
-rc/h
g(x)
,k,
(-l)*"sin(Kw/h)
w + kh
(-D
I
k=-o
I
sin( rcw/h)
w + kh
g(kh) sinc^W ^
ikhx
k e
(- 1 )
w+kh
.
I
g (x)eikhxdx
14
Hence for
f £
B ( h ) , f(w)
= C(f,h)(w)
for all w £ p.
that this is a stronger result than originally
initial quest
was for
a class
of functions
Whittaker series was exact on R.
is derived from Theorem (2.16)
sought.
Note
The
such that the
An even stronger statement
and the identity
I, if A = k
(2.18)
I J
S(k,h)(t)S(A,h)(t)dt =
0,
if A ^ k
that is,
Theorem
(2.19):
The set |
orthonormal set in B(h)
Unfortunately the
S (k ,h) |
[10].
set B(h)
is extremely restrictive and
some relaxation is necessary if (2.5)
interpolatory tool.
here called
BP(Dg ),
approximation to
is a complete
is used as a practical
M c N a m e e , et. a l . [10]
where
C(f,h)
f is very good.
is
identifies a s e t ,
not
exact
In particular,
but its
the domain
of analyticity for B P ( D g ) is D g .
Definition
(2.20):
Dg = (z: z = x +
Figure 4:
iy,
The Domain Dg
|y| < d £ R, d > 0}
15
The class BP(Dg)
Definition
Is specified by Definition
(2.21):
Bp (Dg)
is the
(2.21).
set of functions f such
that
(2.22)
f is analytic in D g ,
(2.23)
J
d
If (t+iy) Idy = 0(|t|a ) as t -> ±® where 0 < a < I,
-d
and for p = I or. 2
.
Np (f ,D o ) = Iim
y-»d
00
„
r ■ If(t + iy)Ip dt
J
.
“
00
.
(2.24)
1/p"
iy)Ipdt
< OS .
The exact form of the error is given
Theorem
(2.25):
by Theorem (2.25).
If f E Bp (Dg ) .then £ (f) (x) = f (x) -
C (f ,h ) (x) where
san( jrx/n;
e ( f ) (x)
J
_______ f(t-id)_________
(t-x-id) sin[ it(t-id) /h]
(2.26)
f(t+id)_________
(t-x+id) sin[ jt (t+id) /h]
Moreover
if f E B 1 (Ds ) '5 B(Ds ) then
N 1 (f,Dg )
(2.27)
e (^) H00 - 2,red sinh( icd/h)
N ( f ,Ds )
2rcd sinh( Tcd/h)
16
while if f G B 2 (Dg ) then
N 2 (f ,D g )
Ile (f) Ilro < ------ =
---------2 VE3 sinh(Kd/h)
(2.28)
and
N 2 (f,Dg )
llc(f) H2
(2.29)
For a
proof see
s i n h (Kd/h)
Stenger
[11].
Worth
error statement of Theorem (2.25)
line.
Theorem
However,
(2.16),
the original goal
line and Theorem
Of far
(2.25)
IIs(T)IIo3 2
B2 (Dg ) .
is that the
is valid only
on the real
applies to the complex plane.
was
approximating
on
the real
certainly satisfies that.
greater interest
from (2.27) , (2.28),
hence,
recall,
noting,
is the order statement derived
and (2.29).
As h
-> 0
sinh(jcd/h)
-» ® ;
0 independent of whether f E B 1 (Dg ) or
M o r e o v e r , the rate of convergence is governed by
sinh (Kd/h) ; i.e.,
l/sinh( rcd/h) = 0( exp (-Kd/h) ) as h
0.
Although the exponential convergence rate is attractive,
to be of practical importance
(2.5)
is
truncated.
it
must
be
maintained when
Denote the truncated Whittaker series
for a function f by
^
(2.30)
CM > N (f,h)(x)
=
Y
J
/x - kh\
f (kh)sinc^— ---
k=-M
where it is assumed C (f ,h ) (x) converges.
Theorem (2.31):
If f G BP(Dg ) for
p =
I or
2, d
there exist positive constants a and (3 such that
>0
and
17
(2.32)
e«x
if x 6 (-=,0)
S - I3x
if x E [0,® )
|f(x)I < L
then choosing
(2.33)
N =
a
M + I
P
and
h = (K d / (a M ) )%
(2.34)
gives
(2.35)
Hf - CM / N (f ,h)||oo < C M* e “ (JtdaM)^
where C is a constant dependent on f .
Proof:
that
From Theorem
(2.25)
there exists a constant L 1 such
If (x) - C (f ,h) (x) I < L1S-icdZ*1
the triangle inequality and
for
all x
J
|f (-kh) I +
k=M+l
If (k h ) I
k=N+l
k=M+l
£ B- I3kh
k=N+l
e-aMhe- PNh
----- + —
< L1B-jcdZh + L
Now if N and h are defined by (2.33) and (2.34),
S Lie- ( ^ M ) X
J
J S“ akh +
< L 1B-jcdZh + L
- CM ,,(f,h)(x)|
Using
(2.32),
|f(x) - CM f N (f,h) (x) I < L l6- jcdZh +
K(X)
E R.
+
+
then
e — {xdaM)X
_-(JtdaM)^
= C M7* e
The choice
errors.
of N
and h
is dictated by balancing asymptotic
The truncation errors for the lower and upper sums
18
are of orders 0
(
a
n
d
Ofe- ^ * 1), respectively.
these order statements gives N =
convenience to
Using (2.33)
guarantee N is an integer.
is deduced by equating
and the
aM/|i.
the truncation
Equating
is a
The choice for h
error order 0(e- a M h )
order of the interpolation error 0
(
)
.
Hence,
truncating sums for computational feasibility need not be at
the expense
of the exponential convergence rate.
carries over during the
development of
This rate
the sine quadrature
rule.
Quadrature on
The
interpolation
(-=,«)
results
just
reviewed
are
the
groundwork for the derivation of the sine quadrature rule on
the
real
line.
As
with
interpolation,
formula involves infinite s u m s ;
deducing the
error caused
line.
The use
a
primary
by truncation.
numerically practical means of
the real
hence,
the quadrature
task is
This leads to a
estimating an
integral over
of conformal maps generalizes the
sine quadrature to alternative c u r v e s .
A few preliminaries
are
necessary
before
stating the
quadrature theorem analogous to Theorem (2.25).
/x - kh\
sincf— ---- Idx =
Lemma (2.36)
.iax
Lemma
(2.37)
f
x - z,
I
,
h > 0, k 6 Z
e iaz0,
Im(Z0 ) > 0
0
Im(Z0 ) < 0
2ici
,
.
19
Lemmas
(2.36) and (2.37) are proved using standard contour
integration.
Lemma (2.38):
If f G L 1 (R) then J C ( f ,h)(x)dx = h
£
f(kh)
k=-=
where h > 0 and k E Z.
Proof:
By the integral test
£
|f(kh)|
converges.
k=-®
Since
/x — kh\
If(kh)sinc^—
---j( < |f(kh)|
,
/x
x —
- kh
f (kh)sincf— -
J
k=-<»
converges absolutely and,
M - t e s t , uniformly.
!
C (f,h)(x)dx
t h e r e f o r e , by the Weierstrass
Thus
f
I
-Co
f(kh)sinc(^
^ ^dx
k=-
^
f(kh) J
)dx
—0
Ic--OO
= h
sinc(iL_— —
2
f(kh)
k=-oo
These three lemmas along
with.
Theorem
background to prdve the following.
(2.25)
provide the
20
Theorem
(2.40)
(2.39):
J
—
If f G B(Dg ) s B 1 (Dg ) then
f (x )dx = h
CO
£
—
f (kh) + r?(f)
—
OT
where
(2.41)
I
f?(f)
c (f )(x )dx
and
e -jcd/h N (ffDs)
(2.42)
|i?(fH
Here s(f)(x)
S
2sinh( Jtd/h)
and N( f , D g )
are defined
in (2.26) and
(with p = I), respectively.
Proof
f
s(f) (x)dx
I
f (x )dx - J
C (f ,h )(x )dx
J
f (x )dx - h
£
—
f(kh)
Ic=-CO
CO
From the left hand side, using (2.26)
oo
17(f) s J
J
J
E(f)(x)dX
sin(rex/h)
2rci
J
f (t-id)dt
(t-id-x) sin( it(t-id) /h)
f (t+id)dt
(t+id-x) sin( jc(t+id) /h)
(2.24)
21
irct/h _ f (t-id)e -I irt/h
00 f (t+id)e
(t+id)e:L,Tt/n
J sin( Jt(t+id) /h)
sln( jc (t-id)/h)
e-n:d/h
21
where F u b i n i 1s Theorem and
Z0
=
t
±
id
give
expression leads to
(2.37) with a = ic/h and
the last equality.
(2.42).
The result of Theorem (2.39)
the familiar
Bounding the last
trapezoidal rule
the rate of convergence for
is that sine
on the
(2.40)
quadrature is
real line.
However,
is 0 (e -2,rd/h ) rather than
the usual 0 (h2 ) that is associated with the trapezoidal rule
when f" is b o u n d e d .
restriction
f
G
Sine
B(Dg )
function properties
which,
in.
lead to the
turn, accounts for the
vastly accelerated convergence rate.
As with interpolation,
truncating
the
sine
numerical
practicality calls for
q u a d r a t u r e , n e e ' trapezoidal,
series.
Define the truncated trapezoidal series
(2.43)
TM N (f,h)
N
5 h £
f(kh)
, h > 0 .
k=-M
Analogous to Theorem (2.31).is Theorem (2.44).
Theorem (2.44):
for some
positive constants
as in (2.33) and
(2.45)
If f G B ( D g ), d > 0 and f
(2.34) gives
satisfies
(2.32)
a and |3, then choosing N and h
22
< K e_2ird/h + - e-aMh + - e“ pNh
1
a
.
p .
< c e-(2judaM)^
where K 1 , K, and C are c o n s t a n t s .
The
proof
(2.31);
[6].
follows
in
a
like
fashion to that of Theorem
for greater detail see Stenger
Once again
selections for
asymptotically
balancing
[12],
N and
[13] and Lund
h are
errors.
motivated by
Finally,
note
the
increased rate of convergence for quadrature,
0 (e- (2)tdaM)
^ versus interpolation,
0(e_ ^ d a M ^ ) .
Interpolation and Quadrature on Alternative Arcs
The preceeding results hold
useful
in
the
setting
of
for
x
G
building
To be
a
Sinc-Galerkin
approximate for the solution of a differential equation,
results
Stenger
must
be
generalized
to
[5] shows that conformal maps
extension.
alternative
the
inte r v a l s .
provide the
means of
His discussion is briefly outlined here with the
goal in mind the statement of the quadrature rule.
Let D be a simply
Definition
(2.20).
a conformal map X :
(2.46)
and
connected
domain
and
Dg
be
as in
Given a,b G 3D such that a ^ b, there is
D
Dg (see Figure, 5) satisfying
V(R) = X - 1 (R) = Y
Figure 5:
The Conformal Map X.
With respect to D, the following definition is similar to
( 2 .2 1 ) .
Definition
(2.48):
Let B(D) be the class of functions such
that f is holomorphic on D;
(2.49)
J
If (w )dw I = 0(|t|a ) as t -> <*>
V (t+L)
where
(2.50)
a G [0,1) and L =
N( f ,D) E Iim inf
CCDC-»9D
(iy:
|y| < d)
f |f(w)dw|
^
; and
< =°
where C is a simple closed curve in D.
Rather
rules
for
than
D
developing
from
interpolation
and
quadrature
s c r a t c h , two theorems provide a natural
link to the past w o r k .
24
Theorem (2.51):
If
V is
a conformal
simply connected domain D and if
f G
map of
D g onto the
B ( D ) z then
F G B(Dg )
where
(2.52)
5 f (Y{w))Y-'(w)
F(w)
.
Also
Theorem
(2.53):
If
X:
D -> D g is a c o n f o r m a l , one-to-one
and onto mapping then V = X -1 is c o n f o r m a l .
Stenger
is a
[13] remarks on Theorem (2.51)
standard complex
while Theorem (2.53)
analysis r e s u l t .
Finally a revised
version of the exponential decay property
Definition
(2.54):
Let f
satisfying the
conditions
-c G Y
=
=
V(R)
exponentially
G B(D),
X
of Theorem
(2.53),
X -1 (R) •
with
(2.32)
respect
Then
to
be
f/X'
X
is needed.
a conformal map
and
is said
if there exist
to decay
positive
constants K, a and P such that
f (X)
(2.55)
exp (-a |X(T:)|)
, T G Yl
< K
X ' (T)
e x p (-pI X (t ) I) , T G y r
where
Yl =
{z:
z
E
f(
(-==,0) ) }
(2.56)
Y r = {z:
Theorem
(2.51)
Z G V ( [ 0 , » ) )}.
and Definition
(2.54)
suggest
that the
conformal map is incorporated into the previous theorems and
25
definitions quite
easily.
The interpolation arid quadrature
rules that follow support this notion.
Theorem (2.57) :
hypotheses of
Let f G
B(D) and
Theorem (2.53).
= Y (k h ),
(2.33) and
D
Dg satisfy the
F u r t h e r , suppose f / X 1 decays
exponentially with respect to X.
and zk
X:
If z E y
= Y(R)
= X - 1 (R)
k E Z, h > 0, then selecting N and h as in
(2.34),
respectively,
N
gives
f (Zlc)
X (z )-kh
h
(2.58)
< K e~(%daM)%
where K is a constant independent of z.
Note that to interpolate f rather than f / X 1, simply let
fX 1
and
substitute
hypotheses of
into
(2.58)
Theorem (2.57).
F =
assuming F satisfies the
Equation
(2.58)
shows that
the rate of convergence for sine interpolation on a curve y
remains 0 (e~ (^dxxM) ^ )
Similarly,
the
rate of convergence
for the sine quadrature rule is u n c h a n g e d .
Theorem (2.59):
(a)
IffE
(2.57),
B(D) and X , Y , z, and Zjc are as in Theorem.
then for h sufficiently small
26
(2.60)
I
»
^
f (z)dz - h
f (zk )
'(Zk )
N(f'D) ^ - 2 ?rd/h
1-e"
k=-=
B K 2 e-2xd/h
(b)
Further,
if f / X 1 decays exponentially with respect
to x then
(2.61)
J
I
f (z )dz - h
f(=k)
X 1 (Zk)
< K 9 e-2*d/h + K e -«Mh
2
a
k=-M
+ E e“ PNh
P
(c)
Finally,
if N and h satisfy (2.33) and
respectively,
(2.34),
then for some constant 0 depending
on f
(2.62)
K 0 e“ 2icd/h + - e_aMh + - e” PNh < C e™ <2jrd(xM)?i
2
. a
.
P
Steriger [12]
provides proofs
for centered
s u m s , that
these to
handle M # N.
rationale for
the
use
suitable assumptions
of Theorems
is, M
=N.
(2.57) and (2.59)
Lund
[6] expands on
Both of their proofs illustrate the
of
conformal
maps;
namely, under
the maps are a means of transferring a
standard set of results from Dg to various domains D.
significant,
.
Most
is that conformality preserves the e r r o r .
Sinc-Galerkin Method
\
The previous
discussion is
\
the background necessary to
\
approach the numerical solution of differential^equations
27
via
the
work,
Sinc-Galerkin
it is sufficient
method.
For the purposes of this
to examine
the method
specific class of differential equations.
the second-order,
applied to a
Hence,
consider
self-adjoint boundary value problem
Lf(x)
s f"(x)
+ v(x)f(x)
= o (x)
, a < x < b
(2.63)
f (a) = f(b) = 0
(b,0)} C SD
n
O
*<
{(a,0),
1C
.Il
O
<
in
(2.64)
(U
A
X
A
O'
Let D be a simply connected domain with
and
(2.65)
With regard
to D ,
the solution of
the Sinc-Galerkin
(2.63)
set of basis functions
(2.66)
where X :
(2.53).
L
.
is summarized as follows.
(Sj.
)
S i (X) s S (i , h) o x(x)
D
method to approximate
by
, h > 0
Dg is described in (2.46),
See Figure 5.
Define the
(2.47),
and Theorem
Next define the approximate solution
by
A
(2.67)
fm (x) s
N
£
fiSi (x)
, m = M + N + I .
i=-M
To determine the unknown coefficients,
f^ , orthogonalize the
residual with respect to the basis e l e m e n t s ; that is,
0 = (Lfm - o, Sp )
(2.68)
= . ( fm " , Sp ) + (vfm - a ,
Sp ) , -M < p < N .
28
Here,
the inner product
is
b
(u,v) = J u(x)v(x)w(x)dx
(2.69)
a
where w
is a
weight function
yet to
be specified and the
quadrature rule used to evaluate the inner product is
(2.61).
The resulting discrete linear system is
solved for
fi , -M < i < N.
A general
and Finder
by which
review of
[3].
ease
Their discussion
to judge
(i) the ease of
of
constructing the
evaluating
regard
to
demonstrated by
on the
it is
function leads
the
and
the
inner
(iv)
the
shown
that
an
(ii) the
p r o d u c t ; (iii) the ease of
accuracy
Sinc-Galerkin
elements and
These include:
basis elements;
of
method,
(2.66) while the second
sine basis
the third,
includes several criteria
the quality of a method.
solving the system;
With
Galerkin methods is found in Botha
the method.
the
is shown
first is
to depend
sine quadrature rule.
adroit
choice
to a system which is easily solved.
For
of weight
Lastly,
the accuracy follows directly from Theorem (2.59).
Stenger
[5] thoroughly discusses W =
introduces w = 1/(X')%.
1 / X 1 and
Lund [6]
S t e n g e r 's choice of weight function
handles regular singular problems but yields
a nonsymmetric
linear
results*
system.
Lund's
weight
function
in
a
symmetric linear system while possibly restricting the class
of functions
to which
the method applies.
The development
of the discrete system for a general weight motivates the
29
choice w = I / ( X 1
as well as addresses the questions raised
in the previous p a r a g r a p h .
Continuing with
inner product
(2.68),
for
-M <
p <
N integrate the
(f",S ) by parts twice to get
b
O = J
f " (x)Sp(x)w(x)dx
a
+
[V (K )f (x)
- O ( X ) ]S p ( X ) W ( X ) d x
1
+ J
J
f (x){[Sp (X)W(X)]" + v (X)Sp (X)W(X)}dx
o (x)Sp (x)w(x)dx
(2.70)
b
BT + f f (x)
r d2
M
-S (p,h)
+ S(p,h)
I "
S(p,h)
o X(X)
o x(x)
( X '(x))^w(x)
(X"(x)w(x) + 2X'(x)w'(x))
o X(x)(w"(x)
+ v(x)w(x))
(x)S (p,h) e X (X )W (X )dx
where BT is the boundary term
b
{ f 1(x)Sp (x)w(x)
- f (x)[Sp (x)w(x ) ] 1 \
(2.71)
BT =
For now,
BT is assumed to vanish.
a
•
The exact assumptions
30
governing BT
=0
are discussed
conformal maps are
(2.61)
to (2.70)
used.
To
in Chapter 3 when specific
apply
the
sine quadrature
several suppositions are necessary.
First,
f [SpW]" satisfies the hypotheses of Theorem (2.59).
fvSp w and
a S p W are in B ( D ) .
It is unnecessary for f vSpw / X 1
and oS w / X ' to decay exponentially with respect to
X.
This
(2.60)
for g
evaluation.
That
P
is because
for g
6 B(D),
the quadrature rule
integrated against the sine
Second,
yields point
is,
(2.72)
J
g(x)Sp(x)dx = g(ph)
+ 0 (e-n:d//h) .
a
To proceed,
the following three identities are u s e f u l .
I , p = i
(2.73)
S (p ,h)
o x(x)
=
6
(0 )
0 , p 7* i
X=X,
(2.74)
hi—
= 6
S (p,h) oX (x)
0
(I)
x=x.
(-l)l-P
, P t4 !
. i-P
and
S (p ,h) o X(x)
X=X .
LdX'
- jc2/3
(2.75)
6(2 )
Pi
, P = i
-2(-l)i-p
, P T5 i
(i - P )'
Applying the quadrature rule results to (2.70) yields
31
N
I
f (X i )
i=-M
I
~2
&p V
Lh
X ' (x I^w (x I)
M x / X u (Xi )
+ h 6P i l ( F H I T
w(Xi) + 2w m x I i
(2.76)
.
+
-
p i
X X
?
6(0)
I
1
v l x I lwfxI 1
(X i )
X
1
0txI lwtV
p i
X
1
(X i )
. + o,e-(KdnM,^,
( X i )
i=-M
An equivalent matrix formulation is
—
I (2 ) D(X'w)
h2
+ - I (1) D "I'" + 2w'"J
Lx'
J
h
(2.77)
+
D
w"
+
VW
X'
where
(2.78)
? -
^-M+1'
^N-I'
^ '
GT(X-M)
g ( x -M+l)
(2.79)
D(g)
G(Xw)
mxm
32
I (2)
6(2)
Pi
2
2(- I )m
(BI-I) 2
2
- JC
2
2 ( - D m- 1
3
72
(m - 2 ) 2
(2.80)
2(- I )m
2(- I )m 1
(m- 1 )2
(m- 2)2
mxm
and
1(11
■ t t ¥ )
i.
0
-1
I
0
1
2
-
( - D m' 1
m-1
( - D m"2
1
m-2
(2.81)
(_l)m -l
- ---- --_
m-1
For an arbitrary weight
solve
(2.77).
(_1 )m-2
- ---- -—
m— 2
function it
. . .
0
is unclear
how to
In p a r t i c u l a r , the skew-symmetric nature of
I (1 ) causes difficulties.
It can be argued that as h -»
I (2 ) is the dominant matrix in the system.
0
This is somewhat
satisfying in that l(2 ) is a symmetric, negative definite
33
Toeplitz matrix
Lund's
(2.82)
whose spectra lies in the interval
( - K 2 ,0)
[6] notion is to consider
X"
— w + 2w' = 0
X'
or equivalently
(2-83)
X"
—2w'
r
" "V”
*
Integrating both sides of
(2.83) yields w =
1/(X')% and the
system now simplifies to
—
I (2) D ( (X ')^) +, D
(X')%/ X '
(X' )*
(2.84)
= D [ o / ( X ' ) 3 / 2 ]f
.
Multiplying through by D(X') and factoring gives
(2.85)
At
= D (o / (X 1)%)I
where
(2.86)
A = D (X ') — l(2 ) + D
I
D(X')
+
(X')*7(X')3/2
(X ')•'
and
(2.87)
f = D(1/(X'
The matrix
.
A is a real symmetric,
negative definite matrix;
h e n c e , there exists an orthogonal matrix Q such that
(2.88)
A=
QAQT
where A is the diagonal
matrix
Solving for t ,
(2.89)
t
= QA -1Q t D (o/(X')^)?
of
the
eigenvalues
of A.
34
in turn, "t is recovered via
and,
(2.90)
? = D ( ( X ')%)# .
In review,
a symmetric coefficient matrix
adjoint problem
(2.1)
The class of singular
appears to
[6].
proof
is attained by selecting w = I / (X 1) ^ .
problems
that
w
=
1/(X')% handles
be somewhat more restricted than if w = 1 / X 1
H o w e v e r , this statement is dependent on
in
for the self-
(6].
singular problem,
When
w
the method of
= 1/(X')% is known to apply to a
the rate of convergence remains
0 (exp(-K'fH)) where 2N + I basis functions are used.
35
CHAPTER 3
THE SINC-GALERKIN METHOD FOR THE WAVE EQUATION
Various
numerical
methods
for solving general second-
order hyperbolic problems can be found in the literature
[3].
Usually a
scheme is
first developed for the classic
vibrating string problem
L u ( x , t) E u t t (x,t)
- ux x (x,t) = f (x ,t )
(x ,t ) 6 (0,1 )x (0, <»)
(3.1)
Once
u(o, t ) = u(i ,t ) = o
t e to,a.)
u(x,0)
x E [0,1]
this
generalizing
= U t (x,0) = 0
model
problem
routes
are
adaptations
to
nonlinearities,
is
investigated,
possible.
handle
Among
.
several
them
are
nonconstant
coefficients,
or higher space d i m e n s i o n s .
With regard to
the Sinc-Galerkin
method,
space dimensions.
In this chapter the discrete linear Sinc-
Galerkin system for
it a
(3.1)
natural extension
the
present work examines higher
is derived in some detail and from
to the
linear systems
three space dimensions is exhibited.
for two and
The actual solution of
the systems is postponed until Chapter 4.
A common
discretization
coefficients.
procedure
of
the
for
solving
spatial
(3.1)
is
a Galerkin
domain with time dependent
This gives a system of ordinary differential
36
equations typically
solvers)
[14].
necessity to
solved by difference techniques
Drawbacks
of
this
artificially truncate
fact that the numerical solution is
time
grid.
In
the
include
the
the time domain and the
valid only
work
implements a Galerkin scheme in time as well as space.
The
functions
the
of
the method is largely due to the choice of basis
e l e m e n t s ; these basis elements
intervals
technique
on a finite
this
success of
contrast,
method
(O.D.E.
composed
with
suitable
(0,1) and (0,~ ).
means
for
building
are tensor
The
an
products of sine
conformal
maps for the
conformal map
for
(0,«)
is
approximate solution to (3.1)
valid on the infinite time domain.
The structure of the
discrete
computationally efficient
result of
the
for at
identities
quadrature rule
(2.61),
least two
(2.73)
-
system is
reasons.
(2.75)
and
As a
the sine
the coefficients of the unknowns and
the constant terms are easily
property is
Sinc-Galerkin
evaluated.
Note,
independent of the weight function.
that this
A property
which is intimately connected to the selection of the weight
is the symmetry of the system.
shown that a
generalization
symmetric linear
system.
discrete system are given.
to implement
i.e.
In the present chapter it is
of
w
=
Two matrix
The
I / ( X 1)^
leads
to a
formulations for the
more convenient formulation
is dependent on the computational environment,
the available computer h a r d w a r e .
37
To
determine
solution of
the
Sinc-Galerkin
(3.1) the
fundamental steps
remain u n c h a n g e d .
First,
then
these
basis
The
unknown
using
solution.
approximate are
with respect
used in
functions
define
coefficients
((x ,t ):
0 <
intervals
appearing
the
The orthogonalization
by means of
(2.61).
I, 0
< t
and (0,=>)
products are f o r m e d .
(3.2)
in
basis functions on the region
x <
(0,1)
Chapter 2
an approximate
is defined with a weight function and evaluated
To construct
the
orthogonalizing the residual
basis elements.
the quadrature rule
for
select a set of basis functions,
determined by
to the
approximate
0(z ) = A n
< “ },
basis functions
are built
So to begin,
on the
and then their tensor
define the conformal maps
1 - z
and
(3.3)
T(w) = A n ( W )
The map 0 carries the eye-shaped region
(3.4)
D- =
onto the
infinite
Similarly,
(3.5)
onto Dg .
Z = X + iy:
strip
arg
Dg
I —
given
< d < jr/2
z
by
Definition
the map T carries the infinite wedge
Dw = (w = t + is:
Iarg(w)|
< d < rc/2}
These regions are depicted in Figure 6.
(2.20).
38
Figure 6:
The Regions Dfi, Dw , and Dg .
The compositions
(3.6)
S 1 (X) = S(i,hx ) o 0(x )
and
(3.7)
S j (t) = S(j,ht ) o T(t)
define the
basis elements for
(see Figure 7 b e l o w ) .
the mesh
(0,1) and (0,°°), respectively
The "mesh sizes" h x and
sizes in Dg for the uniform grids
(Ichx ), -a> < k < C0, and (pht ) , -o» <
points xk 6 (0,1)
p <
in Dg and tp G (0,»)
<=.
and
Xjc = 0 1 (khx ) = e
khx
/(I + e
khx
The sine grid-
in Dw are the inverse
images of the equispaced g r i d s ; that is,
(3.8)
h t represent
)
39
Pht
(3.9)
tp = T “ 1 (pht )
S(O1I) o <j)(x)
Figure 7:
S(0,1) o T(t)
The Basis Functions S q (X) and S*(t).
The fully Galerkin approximate is now defined by
(3-1°)
X
^
t Ix '!) '
™x
“t
I'
I
I=-Mx
U ljS 1(X)Sj It)
J=-Mt
mX
=
MX
+
NX
+
1
mt = M t + Nt + I
The inner product used to orthogonalize the residual is
(3.11)
(u ,v )
J J
0
u(x,t)v(x,t)
0
dxdt
[0'(x)T'(t)]*
and may be viewed as the double integral analogue of
with w
= 1/(X')%.
Galerkin system for
(2.69)
Direct development of the discrete Sinc(3.1)
is obtained via
40
-M
(3.12)
0 = (Lu
X
However,
this
< k < N
- f, Sk S ) ,
•
T
-Mt < p < N t
procedure
obscures
the important parameter
selections needed to implement the m e t h o d .
The development
of
one-dimensional
the
Sinc-Galerkin
systems
for
the
8
A
A
= r(t)
O
u"(t)
rt
problems
(3.13)
u(0)
= u ’(0) = 0
and
0 < x < I
u" (x) = S(x)
(3.14)
u(0)
yields all of
problem
while
= u(l) = 0
the
sine
clearly
matrices
,for
the two-dimensional
disclosing the parameter selections
just mentioned.
Beginning with
in
the
orthogonalization
detail).
(2.60)
(3.13), perform two integrations by parts
Once
this
is
of
done,
the residual
apply
the
(see
(2.70)
quadrature rule
in Theorem (2.59) with X replaced by T to get
[u"(t) - r(t)]{S(p,ht ) o T(t)(T'(t))-%}dt
u ( t ) {S(p,ht ) o T(t)(f'(t))-*}"
(3.15)
for
r(t){S(p,ht ) , T(t)(T'(t))-*}dt
dt
41
= ht
£
U ( t n.)f I r 6,(2)
I 6p ° ’ )
J=-C
r(tp )
- h4
3/2 + 1U + Jr
(T'(tp ))
The second equality in (3.15) assumes the boundary term
( S(PfJK)
o T(t) (T' (t) ) « u' (t)
(3.16)
[S(p,ht ) o T ( t ) (T'(t))
vanishes.
' u (t )}
|0
Using the definitions of T (3.3) and S( p , h t ) °
this is true as long as
(3.17)
Iim u'(t)
-Tt/An (t ) = 0 ,
(3.18)
Iim u ( t ) / J T = 0 ,
t->o+
and
(3.19)
Iim u(t)/(Jt
f+00
« A n (t )) = 0 .
With regard to the last equality in (3.15),
defined by
(3.20)
the nodes tp are
(3.9) and the identity
(T'(t))™3/2
is utilized.
[(T'(t))
%]" = -1/4
The integrals Iu and Ir are explicitly defined
in [15] and represent the exact error terms in (2.60).
Theorem (2.59) , I11 vanishes with order 0(e-,cd/h) if
u
(3.21)
u(z) [S(Pzht ) o T(z)(T'(z))~"%]
E B(Dw )
while the same statement holds for Ir if
(3.22)
r(t)(S(p,ht ) o T(t)(T'(t))-%)
E B(Dw ) .
From
42
to truncate the infinite sum note that
u ( t ) (S(Pzht )
o
t ( t ) (t'(t))“^>" behaves like u ( t ) (?•(t))3 ^ 2
near t = 0 and t = ®.
Hence,
condition (2.55) simplifies to
tY
(3.23)
|u(t)(?■(t))%|
, t 6 (0,1)
< K
t “ 6 , t G [I,*)
for positive constants K,
(3.18)
and
for (3.16)
(3.19)
the
to vanish
quadrature rule
0 = h
t.
(2.61)
I
v, and
6.
Since
(3.23)
implies
only additional assumption necessary
is
(3.17).
Applying
the truncated
to (3.15) yields
u l t J lI r 1 = 6P J 1
! - 4J S V ' jtJ 1!*
J=-Mt
(3.24)
r(t )
- ht --------— — + 0 {exp (-JtdZht ) )
(t'(tp ))3/2
+ 0(e x p (-YMtht )) + 0(exp(-6Nth t ))
If ht and Nt are chosen by
(3.25)
ht = (JtdZ(YMt ) )%
and
I
(3.26)
then the
errors are
asymptotically balanced
and the final
linear system is
I ,(0)
4 6Pd
(T'(tj))% u(tj)
(3.27)
- r(tp )Z(T'-(tp ) ) 3^ 2 = 0 ( e x p { - (JtdYMt )^ ) ) .
Therefore,
if
43
Nt
£
a
(3.28)
Um ^ (t) s
UjS(Jzht ) o T(t)
, mt = Mt + Nt + I
J=-Mt
is an assumed approximate solution of
'
(3.13)
then since
-
A
um t (tp) = Up,
is
defined
the discrete Sinc-Galerkin system for
by
replaced by U j .
for
(3.27)
(3.29)
the
side
left-hand
When p = - M t , .. .,Nt
of
(3.13)
,(3.27) With u (t .)
■ J
the matrix formulation
follows from (2.84) and reads
A tD((T')%)u = D((T')“ 3/2 r)3?
where
and the
At is
remaining terms are defined by (2.78) and
a
real
maintain these
symmetric,
definite
matrix.
properties in the discrete system (3.29)
change of variables
(3.31)
negative
(2.79).
To
the
.
v = D((T')-%)u
leads to
(3.32)
A t v = D((T')~% r)f
where the matrix
(3.33)
i
A t = D ( T ' ) A tD(T' ) .
The solution u is found via the procedure outlined in (2.88)
through
(2.90).
Before considering
(3.14) note that the selection h t in
(3.25) asymptotically balances the error terms exp(-jcd/ht )
and exp(-vMth t ) while the selection N t in (3.26) balances
44
the
asymptotic
error
term
errors
e x p (-6Ntht )
exp(-6N^h^)
arises
as
inequality (3.23) which assumes u(t)
infinity.
For
many
problems
and
a
exp(-yM^h^).
consequence
The
of the
decays algebraically at
the
solution
decays
exponentially as
(3.34)
|u(t)(?'(t))%|
In this case,
(3.35)
,
t G [I,-)
.
Lund [6] shows that the' choice
——
h t An
significantly
< K e " 6t
+ I
—
t.
6 M^
th r
reduces
the
size
of
the
discrete systems
solved with no loss of accuracy.
The preceeding discussion applied
parallel development.
(3.2)
(3.14)
follows a
The map T is replaced by the map 0 of
(since 0 is compatible with the interval
is substituted
for ht .
(0,1)) and h x
Again orthogonalizing the residual
and integrating by parts
to
to
twice renders
(3.15) with a boundary term on
an equation similar
(0,1) analogous to (3.16).
To guarantee the boundary term vanishes it is assumed that
(3.36,
Iim
x-»l
x_rj-
“ 5 + U,(X,'E
An(x)
Example 5.4 in Chapter 5
unbounded
as
x
-»
O+
illustrates
yet
discussion is included in
decay condition (2.55)
u ' (x)Vl-x
An(l-x)
(3.36)
that
a
Iu(x ) (0'(x))%|
where
is satisfied.
example.
< L
(I
- x)P
u'
is
Further
The exponential
simplifies to
xa
(3.37)
case
,
,
x E (0,%)
x G [%,!)
45
for
positive
replaced
by
constants
u(0')3'/2
L,
and
a,
0,
and
|3
where
f and X are
respectively.
Hence
the
selections
(3.38)
h x = (rcd/(aMx )
and
(3.39)
I
Nx .=
result in
the balanced asymptotic error rate
0 (exp (-(JtdaMx )^) ) .
T h e r e f o r e , if
a
(3.40)
Umx =
Mt
£
U iS ( I fIix ) o 0 (x)
,
mx = M x + Nx + I
i=-Mx
is
an
assumed
approximate
solution
of
(3.14)
then the
discrete Galerkin system for the (U1 ) is given by
(3.41)
Axw = D ( (0')“*
s)t ■
where the matrix
(3.42)
and
A x = D(0')AxD ( 0 ' )
A x is
the same as A t in (3.30) with h t replaced by h x -
As with A. > A
is symmetric and negative definite.
t
x
when M
x
= N ,
x
D ( 0 1) is
centrosy m m e t r i c .
T o e p l i t z , Ax is centrosymmetric.
Finally,
Further,
Thus since Ax is
the vector
w is
related to the vector of unknowns u by
(3.43)
w = D( (0 ')_J*)u .
The separate
one-dimensional problems serve to identify
the matrices Ax and A t as
selections.
well as
disclosing the parameter
Returning to the two-dimensional hyperbolic
46
problem
(3.1)
and
its
approximate
solution
(3.10),
the
unknown coefficients (u Jj) are found by
orthogonalizing the
residual.
for
One
possible
formulation
the
resulting
discrete Sinc-Galerkin system is
(3.44)
Ax V - VAt = G
,where the matrices A t
and Ax
are identified
in (3.33) and
(3.42).
The mx x mt matrices V and G are defined by
(3.45)
V = D((0')"%)UD((T')-%)
and
(3.46)
G = D((*')-%)FD((T')-%)
where U
and F are the Mtx x mt matrices which consist of the
coefficients
The
form
(u ^j) for
of
(3.44)
(3.10)
suited
to
subsequent use of the
motivation
(3.45)
At .
that
respectively.
the unknowns
the
sine
as a
matrix is more
o r thogonalization
quadrature
transforming
procedure and
rule.
the
The second
unknown matrix U by
introduces the symmetric coefficient matrices
As a
(3.44)
is
f(x^,tj),
has two easily discerned m o t i v a t i o n s .
One is that representing
naturally
and
direct consequence
is easy to
of this symmetry,
solve numerically.
Chapter,
A x and
the system
4 discusses
the solution technique at some l e n g t h .
A
second
alternative
for
posing
the discrete system
occurs when the unknowns are arrayed as a v e c t o r .
In some
s e n s e , representing (u Jj) as a
vector
matrix is
purely
however,
a
notational
matter;
versus
the
a
computational
aspects of storing coefficient matrices and numerically
47
solving the
whether
discrete system
can vary
greatly depending on
{u^ . } is regarded as a vector or matrix.
Two background terms provide the notational machinery to
rewrite the
discrete system
posed as in (3.44) as a system
defined w ith (u^j } as a v e c t o r .
Definition
(3.47);
p x q matrix.
Let A be an m x n matrix and B be a
The Kronecker or tenspr product of A and B is
mp x nq matrix
A ® B H
The
second
I
a iiB
a I2B
.
a 21B
^ 2 2®
•
_a mlB
a m2B
• • • •
term,
.
.
*
.
•
concatenation,
a lnB
a 2nB
•
a mnB
loosely
representing a finite ordered array as a v e c t o r .
concatenation is defined
For
now,
a
precise
for
an
array
definition
subscripts is sufficient.
refers
to
Eventually
with n - s u b s c r i p t s .
when there are one or two
If b = (bj),
I < i <
m, then the
concatenation of b is denoted
(3.48)
Hence,
co(b)
for
s Eb1 , b 2 , ..., b m ]T
a column
vector r, co(r)
= rT .
vector c,
co(c) =
c while
for a row
When B = (bj j ), I < i < m and
I < j < n, then c o (B) is the mn x I vector
48
COfbi l )
cO(b I2 )
(3.49)
CO(B) =
c o <b in>
which may
be regarded
Davis [16] includes a
discussion of
as stacking the columns of a matrix.
more thorough
concatenation and
but slightly different
the Kronecker p r o d u c t .
In
p a r t i c u l a r , Davis defines concatenation as stacking the rows
of a matrix rather than its c o l u m n s .
Besides
the
notational
following theorems ease the
apparatus just introduced,
transition
to
a
the
system whose
unknowns are represented as a v e c t o r .
Theorem
(3.50):
dimension,
If
A
and
B
a and (3 are scalars,
are matrices of identical
then
(3.51)
C O (aA + PB) = aco(A) + Pco(B)
Theorem
(3.52):
Let
A,
X,
and
B
be
matrices
dimensions are compatible with the product A X B .
(3.53)
Proof:
CO(AXB)
= (BT ® A)co(X)
Beginning with A X B , its ij-th element is
(AXB)ij =
m
£
k=l
a jk
n
£
A=I
X jca bAj
Then
whose
49
n
m
bAj
I
=
I
A=I
a Ik xkA
'
k=l
The last line is prec i s e l y ,( B ^ A)co(X).
To apply
form of
(3.44)
(354)
where
Theorems
(3.50)
and
(3.52), a more convenient
is
Ax VIm t - I m x VAt = G
Ig
(q
Concatenating
=
mt ,mx )
(3.54)
is
and
the
q
using
x
q
the
identity matrix.
symmetry
of At , the
discrete system admits the equivalent representation
(3.55)
(Imt 0 Ax - A t @ Im x )co(V) = co(G)
which is referred to as the Kronecker sum form.
The
coefficient matrix
(3.56)
Bi=) = Imt » Ax - A t ® Imx - (Blj) , -Mt < l,j < Mt
has an easily discerned block form.
The square blocks
have dimension Hix x Hix and are given by
(3.57)
where
Bi j = S ( O ) A x - ( A t )l j Imx
(At )^j is the
ij-th element
,
-Mt < l.j < Nt
of the
matrix A t .
The
vectors co(V) and co(G) are related to U and F by
co(V)
= co(D((*')-%)UD((T')-%))
(3.58)
= (D((T')-%)
0 D((0')-%}co(U)
= (b((T')-%)
0 D ((01)
and
(3.59)
From
co(G)
(3.58),
it
is
evident
that
C O (F)
(3.55)
is
the matrix
50
formulation which
arises via a natural ordering of the sine
gridpoints from left to right, bottom
follows
to top
where
(Xi ,tj)
( , tp ) if tj > tp or if tj = tp and X i > xk .
The previous discussion may be generalized to the second
order hyperbolic problems in two and three space dimensions.
Explicitly,
the problems considered are
L u (x ,y ,t ) s u tt - v 2u = f(x,y,t)
(x,y, t ) 6 (0,1 )2x (0, oo)
(3.60)
u|
t 6 [0, 00)
=0
'3(0,I)2
uj
= ut j
It=0
= 0
(x,y) G [0,1]2
It=0
and
Lu(x,y,z,t)
(3.61)
s u tt - 9 2u = f (x,y,z,t)
(x,y,z,t)
E (0, l)3x(0,oo)
u|
=0
t E [0,=»)
lS ( O fI)3
ul
O
all on
CO
n-dimensional
CM
Il
C
i—i
O
closed
variables are
t=0
H 1C
C
(0
r-t
open and
O
where
(x,y , z) E [0,1]3
= 0
ut
=
unit
In particular,
construction
elements
because the
Sinc-Galerkin
respectively,
map 0
basis
(3.2)
approximate
are
cubes.
are the
The spatial
the same interval as a matter of ease
rather than necessity.
of
respectively,
is
on
this simplifies the
each spatial interval
used repetitively.
solutions
Hence the
to (3.60) and
(3.61),
51
(3.62)
A
Nx
u mx ,m y ,m t Cx 'Y '*) = J
N
J
Nt
J
u ijAsijA<*'Y-t I
I=-Mx J=-My A = - M t
and
(3.63)
N
I
N
N
Nt
I
I
I
u I j M s IjkAlx
I=-Mx J=-My k=-Mz A = - M t
zy,z, t)
where
(3.64)
S iJA(x ^Yzt ) = S1 (X)Sj-(y)S^(t)
(3.65)
S i JkA (x ZYzz Zt ) = S i (x)SJ.(y)Sk (z)S^(t)
(3.66)
mq = Mq + Nq + 1
and Sp
,
,
q = x , y rz,t
(P = i/J/k) and S^ are given by
u s u a l , the unknown coefficients
(U^j-
(3.6) and
Z
(3.7).
As
(u I j k A ) are found
or
by orthogonalizing the residual using the weight function
(3.67)
w ( x , y ,t ) = 1 / ( 0 ' (x)0'(y)T'(t))^
for the three-dimensional problem and
(3.68)
w ( x , y , z , t) = l/(0'(x)0'(y)0'(z)T'(t))%
for the four-dimensional problem.
Analogous
selections
problems.
to
are
The
the
two-dimensional
c a s e , the parameter
deduced
from
form
the one-dimensional differential
of
separate
equations in y and z are assumed to be like
(3.14).
The choices for the y parameters are
,
-
(3.69)
and
hy = (TCdZ(SMy ) ) H
-
‘
/
one-dimensional
that of problem
52
(3.70)
+ I
based upon the assumption
v
|u(y)(0'(y))%| < L
(3.71)
ye
(1-y)"
where L ,
$,
and
the z parameters,
(o,%)
y e [%,i)
n are positive constants.
With respect to
an assumption like (3.71) with
f replaced
by u and n replaced by v gives
(3.72)
hz
(red/(pMz j)%
and
+ I
(3.73)
To
list
the
linear
systems
for
<u ijA>
additional notational devices are necessary.
I <
i <
view u(3)
and <u ijkA>
For U = (u^j),
m and I < j < n, U is represented as a matrix.
E (Uij^ ) , I < i < m,
I < j < n, and I < A
To
< p, as
a matrix define
m a t (U(3 )) = mat((Ujj&))
(3.74)
Cco(Uijl), C O ( U ij2),
Conveniently,
concatenating
concatenating the matrix;
c o (U (3 ))
the array
i .e .,
c ° ( (UijA))
C O ( U i Ji)
(3.75)
CO(U1J2 )
.coluijp>J
c o (mat( U 1
))
- c°<u ijP >i
3 ^ is equivalent to
53
A recursive
definition suffices to generalize concatenation
for the n-subscripted array U
I < ij < my,
= (u.-
4 ),
1 I 1 Z - • •-1H
I < j < n, as follows:
c o ( U (n))
c o ^ u I 1I2 . ..in ))
co(uili2 : - - i (n- l , l )
(3.76)
c 0 ^ i 1I2 -■ ■i(n- 1)2 )
C 0 ^U ili2 •••i(n-I)imn ^
In turn, using
(3.76) a recursive formula for the matrix
representation is easily given by
mat(U<n ) ) = mat((uiii2<
(3*77)
s [C° (Ui l i 2 - - - M n - I ) ^ '
.........
Hence,
CO(Uili2--- M n - D Z ^
CO(Uili2---i (n-l)n*n)]
going from m a t (U^n ) ) to co(U^n ^)
the last
means
>in))
to
index of U^n ) .
compactly
coefficients when
involves unraveling
These devices provide a convenient
write
the spatial
the
systems
for
unknown
dimension is greater than or
equal to two.
In two spatial variables,
the system analogous to (3.44)
is
(3.78)
{Imy®Ax +Ay ®Im x }mat(v(3 )) - mat( V < 3 ) )At = m a t ( G (3 ^)
54
where for -Mjf < i < Nx , -My < j < Ny and -Mt < A 5 N^
m a t (V^3 ) ) = mat((v--o))
(3) .
= DyXm a t ( u ( 3 ))D((T') *)
(3.79)
(3.80)
,
Dyx = D ( (0 1(Y))~^) ® D ( ( 0 1(x))~% )
and Ay
is the
same as A x in (3.42) with h x replaced by h y .
The matrix m a t (G (3 ^) is
defined
U(3 >
=
replaced
by
F
just
like
(f (x^,y ^ ,t ^ ))
(XizY j ,tA ) is a sine gridpoint.
m a t (V (3 ^) with
where
Adding
a
the
point
third variable
yields
IImzmy ® » x +
Imz
*
® 4V
A* ®
Imymx>”atlv<4)
)
- m a t ( v < 4 >)At = m a t (G^4 I)
where
<3 -82)
Ipq = Ip * Iq
'
and for -Mx < i < Nx , -My < j < Ny , -Mz < k < Nz and
-Mt < A < N t
mat(v(4))
= mat( ( v i i k A ))
.
(3.83)
= D zyx mat(u(4))D((T')
(3.84)
D
%)
= D((0'(z))"%) ® D ( (0'(y ) ) ~%)
,
® D((0'(x))"%)
.
The matrix Az is Ax with hx replaced by h z , and
(3.85)
m a t (0(4)) = Dzyx m a t ( F ) D ( (?')'%)
where F = (f (Xj.,yj ,Zfe,t'A ) ) .
the coefficient
on the
definitions
the discretization
(3.60) and
of
(3.78) and (3.81)
matrices which multiply m a t (
left represent
operators in
Note that in
mat(v(n ))
Laplacian in n - d i m e n s i o n s .
(3.61),
can
of the Laplacian
respectively.
break
up
^ ), j = 3,4,
the
Alternative
discretized
55
The Kronecker
sum form
for the
discrete system in two
spatial variables is derived by concatenating
(3.78).
That
iS'
c o (g
(3 )) = c o ( m a t (G^3 )))
= c o [(Im y ® A x + A y ® Im x >mat<V<3 >)Imt
- ^ y m xm a t Iv l 3 j IAt]
=' 1Bt ® I1my ® Ax + Ay ® Imx}co(mat(vl3)))
- A tT ® Imymx co(mat(v(3;)))
- [ 1B t ® I1By ® Ax * Ay ® Inlx)
- At ® 1Bymx ] c°lv(3J)
Similarly,
the vectorized version of
(3.81)
^1Hit ® ( ^mzMy ® aX + -Fmz ® Ay ®
1Hix + Az ® ^-mymx^
-
(3.87)
- At ® ! m z m y m x j ^ t V ^ h
By
definition
of
natural ordering:
concatenation,
first
followed by the z (for
t-domain.
is
across the
the
= co(G(4 ))
grid
x-domain,
.
is swept in a
then
the y,
(3.87)), and lastly through the
A closer examination of this structure as well as
the solution is developed in the next chapter.
56
CHAPTER 4
SOLUTION OF THE DISCRETE SINC-GALERKIN SYSTEM
Classical Methods
As
mentioned
schemes for time
in
Chapters
dependent
I
and
partial
3, typical Galerkin
differential equations
use a trial function which is a truncated expansion of basis
functions defined
over the
dependent coefficients.
function for
.
(4.1)
(3.1)
spatial domain
For
instance,
and having time
the form of a trial
is
.
u ( x , t) =
£
c i (t)0^(x)
i=l
where t0*) 2=1 is complete in some
the true solution u ( x , •).
respect to 9 •, I
< j
function space containing
Orthogonalizing the residual with
< N,
leads to
a system
of ordinary
differential equations in time.
The literature abounds with
algorithms to solve
of
this
type
truncated time grid [14],
[17].
In contrast,
system
a discrete,
the Sinc-Galerkin method defines the basis
functions on the entire space-time domain.
of the
on
trial function
The coefficients
are now constants, hence a system of
ordinary differential equations is exchanged for a system of
linear equations.
Linear systems also arise when classic
57
schemes such as finite differences or finite elements are
applied to
(3.1) on a truncated time domain.
established routines exist
these
systems.
finite
differences
numerically
systems
In
for
the
or
feasible
method
is
numerical
more
common
scheme
which
to
approximating solutions of
is
Galerkin
solves
essential
the form
and B is called the
setting the
methods,
a
the
discrete
if
the Sinc-
represent a viable alternative for
(3.1),
(3.60), and (3.61).
Conventional algorithms typically
written in
solution of
light of developed techniques such as
(derived in Chapter 3)
Galerkin
Several well-
solve
linear systems
Bx = b where x is a vector of unknowns
coefficient
matrix.
discrete Sinc-Galerkin
In
systems
the present
(3.55);
(3.86),
and (3.87) have the coefficient matrices
(4.2)
B < 2 > - Imt ® Ax - At ® Iax
,
(4.3)
B O ) = Imt ® (Imy ® Ax 4- Ay ® Imx) - A t ® Imymx
and
(4.4)
,
^
B < 4 > = Imt ® (Imziny ® Ax 4 Imz ® Ay ® Imx
+ Az ® 1Iiiymx * ~ At ® 1HizMiyinx
respectively.
Table I lists the dimensions of B ^ *,
j = 2,3,4, and suggests a first consideration for a feasible
scheme —
machine
storage.
Methods
which
require full
-
storage mode are often impractical in the setting of systems
arising from the numerical solution of
equations.
partial differential
This is certainly true h e r e .
For example,
if a
Table I.
Machine Storage Information for the Matrices
Equation
(3.55)
j
2
Dimension of
Nonzero elements
^
.
mm. x m m
X t
m
X
(m
t'' x
in
+
ni
t
(3.86)
(3.87)
3
4
m m m. x m m m
x y t
x y t
X t
-
I)
'
and
m m m fm + m + m - 2)
x y t\ x
y
z
>
m m m m x m m m m
xyzt
xyzt
m m m m (m + m +
x y z tv x
y
of B(j)
m + m - 3)
z
t
'
Nonzero elements
m m^m
X t t
m m m m
xy t t
m m m m m
x y Z t t
mm
x t
m m m
x y t
m m m m
xyzt
O f B<j)
Unknowns
59
coarse grid
= Itiy
with
order IO4 ;
i.e.,
370,000 are
IO8
= mz
elements
nonzero.
must
2500; however,
the
stored
of which
in this case B ^2 ^ is
of the more than six million elements
represented only 247,500 are nonzero.
to position
be
The problem persists when B^ 2 ) has a
fine grid like Hix = m^ = 50.
2500 x
= mt = 10 is used B ^ 4 ) has
Moreover, with regard
the regularity of the nonzero elements suggests
possibility
of
structure
dependent
algorithms
with
accompanying storage savings.
To put
B ^ 2 ) into
perspective,
consider the coefficient
matrix for the usual centered finite difference scheme.
Assume the order of the Sinc-Galerkin method is 0(e
that the
true solution
When mx = 50,
finite
of
(3.1)
differences
and
decays in
time like e- t .
call
an approximate
for
stepsize h = .02 to have an expected error of
O
0(hi ) =
— V ITLy
0(e
).
Further,
truncated short of t = 8.
in
the
spatial
domain
requires the solution,
the
time domain should not be
This translates to
and
50 gridpoints
400 steps in time.
at each time
system with 148 nonzero entries.
step, of
Iterating
a tridiagonal
For purposes of comparison
to B (2 ^ , these systems can be combined into one large system
with 99,100
nonzero elements.
of nonzero elements
finite
differences
option than the present method.
Example 5.4,
the 0(e
Therefore,
is
based on a count
a
more viable
H o w e v e r , as demonstrated in
convergence for the Sinc-Galerkin
method is maintained when the solution u of
(3.1)
is
60
singular.
Finite
maintaining
differences
0(h2)
have
convergence
no
in
direct
the
analogue
presence
of
singularities.
The
matrices
finite element
occurring
due
solutions of
to
finite
difference or
partial differential equations
are often solved via iterative methods precisely because the
methods can be coded
to take
sparse coefficient matrices
advantage of well-structured,
[4].
discrete Sinc-Galerkin systems
Given the structure of the
(3.55),
(3.86),
and (3.87)
methods such as Jacobi, G a u s s - S e i d e l , and SOR may, with some
modification,
be
practical but
foreseeable modification
taken of the structure
is that
of
is troublesome
unexplored.
A
greater advantage must be
j =
location of nonzero elements.
of
they remain
2,3,4,
than
just the
Storing only nonzero elements
(see Table I).
Finally, with respect
to convergence the success of an iterative method depends on
the spectrum of the iteration matrix.
detailed analysis
be useful.
work
on
providing
In
of the spectrum of
spectrum
the
proof
of
of
B ^ 2 ),
a
, j = 2,3,4, would
Whereas the author has done
the
this direction,
extensive numerical
the
Conjecture
analytic
(4.11)
argument
below remains
elusive.
This
chapter
details
which solve the discrete
Chapter 3.
Both
two
numerically
Sinc-Galerkin
viable methods
systems
derived in
are direct methods that take advantage of
symmetry and block structure to reduce machine storage as
61
well
as
ease
implementation.
The choice of technique is
somewhat dependent on available
example,
computing f a c i l i t i e s .
one algorithm is a modified block Gauss-Jordan
elimination routine.
Although
it is possible to implement
this method on a scalar machine,
vector machine
architecture.
suited to a scalar machine,
less
machine
needing less
author,
For
storage
storage,
inherently suited to
The other algorithm is wellin
than
the
it is
part,
the
because
block
it requires
routine.
scalar machine
Despite
available to the
a Honeywell Level 66, remains too small to implement
the method in three space dimensions when
m„ = m„ = m_ = m+ = 8.
In this instance a
supercomputer is
a necessity due to storage capabilities.
Solution of the Discrete System in One Spatial Variable
The discrete
as f o l l o w s .
Sinc-Galerkin system
Since
Ax
and
At
are
(3.44) may be solved
symmetric
there are
orthogonal matrices Q and P so that
(4.5)
Q 7A xQ = Ax
and
(4.6)
P 1A tP = At
where A , A t. are diagonal matrices containing the
Nx
eigenvalues < (Xx )V i = - M x
respectively.
(4.7)
Z=
Nt
and <<xt >j
If the change of variables
Q tVP
of Ax and At '
62
is made in (3.44)
(4.8)
then the equation takes the form
Ax Z - Z A t = H
where
(4.9)
H =
Q tGP .
The solution of
(4.8),
in component form,
is given by
-Mx < i < Hx
(4.10)
[ (Xx ) i - (Xt )1-] _• = hjj
X 1
X J
J
J
where h ^ j
is the
from (4.10)
(3.44)
Once Z is determined
is recovered via
in turn, using (3.45) gives U.
If (Xx )i
in (4.10)
-Mt < j < N t
ij-th element of H .
the solution V of
V = QZP t and,
,
= (Xt )j for some indices i and j , the equation
is inconsistent.
In the array
of examples listed
in Chapter 5 this matching of eigenvalues never occurs.
The
author believes the following:
Conjecture
a (Ax ) D o ( A t ) = 0 where a(A)
(4.11):
denotes the
spectrum of A.
This is
connected,
so the author believes,
nature of the spectrum of Lu = -u"
infinite domains
(see deBoor
of this has been
proven,
on compact
and Swartz
continuing
to the different
versus s emi­
[18]).
While none
analytic and numerical
work supports the validity of the conjecture.
Verifying
(3.44)
is
nonsingular
the
consistency
equivalent
which,
in
to
of
showing
turn,
the
system posed as in
the
implies
unique solution for the linear equations
matrix
the
is
existence of a
(3.55).
To
63
establish the
Davis
connection a property of tensor products that
[16] states is useful;
(4.12)
that is,
. (A ® B)(C © D) = (AC) ®
(BD)
assuming each product is defined.
Using (4.12)
and
transforming variables in (3.55) via
(4.13)
co(Z)
= (PT ® Q t )c o (V)
yields
(4.14)
{Imt ® Ax - A t 0. InixJco(Z)
= co(H)
where
(4.15)
co(H) = (PT ® QT )c o (G)
Equation
(4.14)
shows that the eigenvalues of B ^ 2 ) are given
by all possible combinations of
-Mt <
j
<
Nt .
Equally
diagonalization of
B^)
(2 ) need
to
is
the
fact that
j.s accomplished by two intermediate
never be stored.
array with respect
(Xx ) ^ - (Xt )j-, -Mx < i < Nx ,
significant
diagonalizations of much smaller
b
.
matrices.
Indeed,
storage
has
As
a result,
if m x - m t , the largest
mx m t
elements.
This
product represents the number of unknown coefficients in the
trial function (3.10)
under
solving
the
(see Table
transformations
(3.44)
and
(3.55),
I). .
Finally,
(4.7) and (4.13)
respectively,
note that
the algorithms
can
be machine
coded identically.
An
alternative
described below.
method
of
solution
for
Consider the transformation
co( Z D ) e CO( Q 7V)
(4.16)
= (Imt® QT )c o (V)
.
(3.55)
is
64 '
Rewriting
(3.55)
in terms of co(ZD ) and multiplying through
by Ijn^ ® Q t yields
(2 )
B£, CO(Zd ) .= CO(Hd )
(4.17)
where
(4.18)
Bll2 I = Imt • Ax - A t • Imjt
and
(4.19)
. C o (Hd ) = c o (Qt G)
(2 )
The matrix
Bd
is
diagonal matrices.
.
a block matrix all of whose blocks are
Explicitly,
if
the
blocks
are called
B D(ij) then they are given by
(4-20)
where
6D(Ij) “ 6Ijl aX - <*t>ij 1Hlx
-Mt 5 i-l 5 N t
(At )i J denotes the ij-th element of A t ,
structure
has
come
diagonalization of
matrix
Ax .
at
Moreover,
less
than
elements
B^2*
(see
techniques work
matrix
diagonal
minor
machine
that
matrices.
and
storage
needed,
Table
extremely well
inversions
expense
of
only
one
the m x x m x symmetric, negative definite
considerably
of
the
This improved
to
I).
the
B^)
is
save the nonzero
Block
elimination
on the system (4.17) as all
multiplications
After
for
are
performed
on
elimination procedure the
solution is recovered from c o (V) = (Imt ® QJ c o (Zd ) = c o (QZd )
from which U is found via U = D((*')%)V D ( ( T ')*).
Although block elimination techniques
performed on scalar machines,
for
B^2 )
the structure of B ^ 2 ) is
may be
65
inherently suited to vectorized computation.
Jordan elimination routine solving
A block Gauss-
(4.17) has
been written
in explicit vector FORTRAN and implemented on the CYBER 205
computer.
Its performance
was consistent with the results
of the algorithm which diagonalizes both Ax and A ^ .
Solution of the Systems in Two and Three Space Variables
The algorithms detailed in
ready
extensions
Galerkin
for
systems
extension
is
previous
application
arising
actually
block techniques.
the
an.
from
to
the
(3.60)
section admit
discrete Sinc-
and
(3.61).
One
assortment of methods utilizing
The second is based on
diagonalizing A„,
Ay , A t , and Az (if it occurs in the system).
The discussion
begins with the second method.
A „ and A_ are
Y
symmetric
z
hence
there
exist orthogonal
matrices R and S such that
(4.21)
R 7A yR = Ay
and
(4.22)
where
S t A zS = Az
A
and
eigenvalues of
and
(3.87)
Az
are
Ay and
may
be
diagonal
Az , respectively.
Equations
Let
-Mx 5 1 5 N x' -M y
(ZijkA)'
C O (Z 1
)
the
(3.86)
5
= (z^j^) and
J5
Ny , -Mz < k < Nz ,
-Mt < A < N t , be defined by
(4.23)
containing
transformed using changes of variables
analogous to (4.13) as follows.
Z(4) =
matrices
(P7 ® R 7 ® Q7 )c o (y(3 ))
66
and
(4.24)
co(z(4))
= (PT ® ST ® R t ® QT )co(V(4V)
where Q and P are given by (4.5) and
Rewriting
(3.86)
in terms of
[1I t ®
(4.6), respectively.
yields
® aX + Ay * Im x ) - A t ® ImymJ c o l Z <3 >)
(4.25)
= (PT ® R t ® Q T ) c o ( G (3))
while
(3.87)
in terms of
4 ^ becomes
[1Jiit ® ^ m zmy® A x + Imz ® Ay ® Imx + Az ® Imymx ^
<4 -2 6 )
- At ®
= (PT ® S T ® R t ® Q ^ ) c o ( G (4?) .
If the
arrays
= (hi;jA ) and H ^ 4 ^ - (HiJkjl) ,
-Mx < i < Nx , -My < j < Ny , -Mz < k < Nz , -Mt < A
< Nt , are
given by
(4.27)
co ( H ( 3 )) = (PT ® R t ® Q T ) c o ( G (3))
and
(4.28)
CO(H ^4 ^) = (P? ® ST ® R T ® QT )c o (G (4 ^ )
then the solutions of
(4.29)
(4.25) and (4.26),
respectively,
are
C(Xx )i + (Xy)j - (Xt )A ]zijA = h ijA
and
(4.30)
As in
C(Xx )i + (Xy )j + (Xz )k - (Xt )A ]zijkA = h ijkA
(4.10),
the
is inconsistent.
possibility exists that
(4.29) or
Again this never happens for
discussed in Chapter 5.
.
(4.30)
the examples
The author believes in the validity
67
of the
analogue of Conjecture
That Is, based on
spatial
domain
the
is
(4.11)
array
compact
of
examples
while
infinite then, with respect
for (4.29)
the
to the
and
(4.30).
tested,
if the
temporal
is semi­
sine discretization the
spectrum of the discretized Laplacian is disjoint from the
spectrum
unknown
of
At-
Assuming
coefficients
U
^
the
in
the
recovered from (4.29) via (4.23)
(4.30)
may be transformed using
give
expansion
and (3.79).
(4.24)
(3.62)
the
are
Similarly,
followed by (3.83)
t:o
appearing in the trial function (3.63).
One item remains to
When solving
completely
structure
allows
the
products
retaining use of three and four
(Xi j k ), l < i < m ,
suppose X
(Si j ), I < i ,j < m; B =
C =
(Ci j ),
specify
the algorithm.
the system as above the matrix-vector products
which occur are highly structured.
A
conjecture is valid,
I
< i,j
5 p.
Taking advantage of that
to
be
determined
subscripts.
while
For instance,
I < j < n, I < k < p;
(by^),
I < i,j < n; and
The product
(C ® B ® A)co(X)
is
defined by
[ (C ® B ® A ) c o (X)]ijk
P
I
(4.31)
I
ckt
t=l
where
[
I
S=I
air xrst
b js
r=l
]ijk indicates the ijk-th element.
A, B , and C are as before;
I < j < n , I < k < p,
Similarly,
if
X = (xi j k A ), I < i < m , .
I < A < q ; and D = Cdij ), I < i,j < q;
then (D' ® C ® B ® A) co (X) is given elementwise via
68
A) CO (X) 1 2
[(D ® C O B ®
q
(4.32)
Hence,
p
^
^Av ^
v=l
t=l
ckt
is
merely
a
recovering the trial
^
s=l
b Js
^
r=l
a Ir xrstv
set
of
function
nested
loops.
coefficients
Likewise,
from
that
the
(when it occurs)
algorithm
is
just
diagonalization
of
the
characteristic
definitive
described.
linear
systems
posed
as
Ax , Ay , At , and Az
Indeed,
diagonalizations are carried out,
when
the numerical
in
(3.86)
(3.81),
respectively.
method for
a
first,
Further,
the numerical
direct
implementation ease,
discussion.
needed.
The
and
(3.87)
(3.87)
represent
occurring in
The
the
diagon a l i z a t i o n s .
The
is established by the preceeding
is
number
the
minimal machine storage
solve
(3.86)
of
is
These
unknown
coefficients
the trial function and, as such,
also give the
minimum array size n e e d e d .
storage,
(3.78) and
it is m xm ym zm t (see Table I).
the
is
the discrete systems
The maximum array size required to
m xm ym t while for
values
second
of
these
the chief attributes of this
solution of
consequence
of the
solution of
indistinguishable from the numerical solution of
are
3 ) and
x s equally simple.
Note
the
m
the code needed to evaluate the vector c o (h (3 ) ) or
co(H^4 ^)
z(4)
n
Hence,
with regard
to machine
it is not possible to do better than this method.
remaining
class
of
algorithms
mentioned as it has yet to be implemented.
is
The
only
briefly
69
distinguishing trait
the matrices A x ,
of the
Ay ,
undiagonalized.
A t,
Here A t
class is
or
Az
that at least one of
(if
will not
it
the change of variables
(4.33)
CO(Z^3 j ) E (Ifflt © R t ® QT )c o (V(3))
(4.34)
(3.86)
b £3 >
remains
be diagonalized.
example,
transforms
occurs)
As an
to
CO (Zj(3 I) = (Imt © RT © Q T )c o (g (3 >)
where
(4.35)
Bjl3 ) is a
b £3 >
block
(Imy ® A x + Ay ® Imx ) - A t ® Imymx
block matrix all
matrices.
a
= Imt ®
of
whose
.
blocks are diagonal
Therefore, one appropriate choice of solution is
Gauss-Jordan elimination
complication arises
is, either the
routine.
However, a
due to the added space dimension.
block
routine
must
be
written
That
to handle
unknowns with three subscripts or the array of unknowns must
be adapted to two subscripts.
routines
used
to
they may not take
solve
In the latter case,
(4.17)
advantage of
the block
could be applied although,
all vectorization possible.
The dilemma is compounded for the system
(4.36)
B ^ 4 ) c o (Z ^ 4 )) = (Imt ® ST 0 R T © QT )C O (G <4 ) )
where
B(4 )
D
- I mt ® ( 1HizItiy ® Ax +
1IItz
® Ay ® 1Hix
(4.37)
+ Az ® 1 JIlymx ) “ A t ® 1 Ulz IIlymx
and
70
(4.38)
CO(ZjJ4 J) = (Imt 0 ST ® R T ® QT )c o (V(4) )
Here the array of unknowns has four subscripts.
To further
cloud the
picture,
the
transformed coefficient matrices are
the
transformation.
In
manner in which the
stored
particular,
may
if
depend on
only
diagonalized the resulting transformations of B (3 )
require more
storage in
Ax
is
and B^4 J
terms of nonzero elements than the
matrices B ^ 3 J and B1J4 J (see Table I).
Diagonalizing the matrices Ax ,
appears the
(3.60),
easier route
and
(3.61)
techniques are
inclusion in
order terms
but
more widely
the
partial
Ay , A ^ ,
and possibly A z
in the setting of problems
the
author
applicable.
differential
and/or nonconstant
believes
(3.1),
that block
For instance,
equation
the
of lower
coefficients may discourage
diagonalization of the entire coefficient matrix.
71
CHAPTER 5
NUMERICAL EXAMPLES OF THE SINC-GALERKIN METHOD
The space-time
and
(3.61)
known
Sinc-Galerkin method
was tested
solutions
on a
exhibit
large class
differing
problems with known solutions
error
evaluation.
numerical success
nine
examples
With
characteristics of the s c h e m e .
nine breaks
for
to
for all
herein
are
In
(3.1),
(3.60),
of problems whose
behaviors.
allows
regard
is evident
reported
for
a
error,
Choosing
more complete
considerable
p r o b l e m s ; h e n c e , the
selected to highlight
addition the
sample of
into three groups of three where within a group
a given characteristic is
illustrated for
problems in one,
two, and three space dimensions.
Examples 5.1
that
is,
- 5.3 and 5.7 - 5.9 display a first trait;
optimal
parameter
selections
for
an
expected
convergence rate result in a discrete system of minimum size
for the Sinc-Galerkin method.
discrete system
is maintained
(see Examples 5.7 - 5.9).
of the
Moreover,
symmetry
of the
for any choice of parameters
A characteristic
common to each
nine examples is that the numerical solution (3.10),
(3.62), or
(3.63)
In particular,
(depending on space
by employing
dimension)
is g l o b a l .
the conformal map ?(t) = An(t)
the approximate solution is valid on the infinite time
72
interval .
Another
significant
Examples 5.4 - 5.6 whose
respective boundaries.
is,
parameter
the analytic
solutions
are
is exhibited by
singular
on their
With respect to implementation (that
se l e c t i o n s ),
these singular
property
the
Sinc-Galerkin
method for
problems proceeds in the same fashion as for
examples.
following (5.7),
Referring to
the rate
the order statements
of convergence
governed by the Lip a class to which
of the method is
the solution belongs;—
not the degree of singularity of the solution.
Perhaps
the
most
Galerkin method,
distinguished
indeed of
feature
spectral methods
the potential exponential convergence rate
(2.1)
this
convergence
is
of the Sincin general,
[7].
established
For problem
analytically.
Specifically,
a direct consequence of Theorem (2.59)
if
basis
2N
+
I
functions
approximate solution
K >
0,
holds
problems.
of
used
to
construct
for
(3.61) and their respective approximates
(3.10),
rate on
exponential
grid.
convergence,
exist.
theory (see
and singular
(3.60),
and
(3.62), and
.An analytic
extension to uniform
in
dimensions
multiple
Arguments
Chapter 2
lead directly
variables.
an
is easy to establish the exponential convergence
the sine
currently
to
analytic
(3.1),
it
respect
both
problems
(3.63),
is that
(2.1) then the order 0(exp(-K>fW) ) ,
uniformly
With
are
is
based
on
does not
analytic function
for the development in one variable)
to problems
Alternatively,
in functions
of several complex
arguments based on the
.
73
development as in Stenger
[5] are complicated by the form of
Green's functions in higher dimensions.
that the
method
hold
exponential convergence
(indicated in the
uniformly
in
The author believes
rate of the Sinc-Galerkin
discussion following
higher
dimensions.
(5.12)) does
Unfortunately,
the
development of analytic tools to verify this convergence has
not kept
pace with the development and numerical testing of
the m e t h o d s .
Since block techniques for the problems in two and three
space
dimensions
reported
are
have
for
the
diagonalizations.
yet
systems
When
compiler.
discretization of
solved
feasible
Honeywell Level 66 computer
FORTRAN-77
to be implemented,
using
For
(3.60) or
the
a
via
code
double,
the results
all
possible
was run on a
precision ANS
large systems arising from the
(3.61)
the
array
of unknowns
exceeds the maximum array size allowed on the scalar machine
mentioned.
code run
H e n c e , these systems were
on a
solved with
the same
CRAY XMP/48 using a single precision CFT.141
compiler.
Single precision on the CRAY XMP/48 is equivalent
to double
precision on
most scalar
CRAY XMP/48 reproduced the
machines.
error results
Indeed,
the
of smaller double
precision Honeywell r u n s .
With
regard
to
problems
maximum absolute error between
in
one space dimension,
the
the numerical approximation,
u i A , and the true solution, u ( x ir tA ) , at the sine gridpoints
was determined and reported as
74
(5.1)
IIE(1 M h ) N - = max lu-o - u (X1 ,to) I
u .
IA
1
where h
is the
stepsize associated with x.
Similarly,
the problems in two and three space dimensions,
for
the maximum
errors are
(5.2)
IIE(2) (h)ll = max IU 1 1 O - u ( x l ,y1,to)|
u
IjA
J
J '
and
(5.3)
IIE(3 Mh)II = max |U 1J1ca - u (X1 ,yj ,zfe,tA ) |
u
ijkA
respectively,
approximates
u (Xf,Yj/zk /t A )•
where
of
u IjA
the
true
the solution u ( x , t) of
*
u IjkA
values
are
in one
numerical
n ( x 1 ,yj,tA )
The notation .ddd-v represents
To implement the method
(5.4)
an<^
and
.ddd x 10- v .
space dimension,
assume
(3.1) satisfies
ju (x ,t ) j < Mx"+%
(I-X)P+^ t y + * e " 6t
for some positive a, |$, v, and 6.
As a consequence of
(5.4)
there exist constants K and L such that
tY
(5.5)
Iu(x ,t ) (T'(t))%|
,
t E (0,1)
< K
e-8t
,
t E [l,co)
and
(5.6)
|u(x,t)(0'(x))%|
hold uniformly for
taken in
(3.14),
light of
(x,t) E (0,1) x
,
x E (0,%)
(l-x)P
,
x E [%,!)
(0,»).
the one-dimensional
respectively,
the approximate
X a
< L
motivate the
These conditions
problems
(3.13) and
parameter selections for
75
(5.7)
A
u
'm^ = M fc + Nt + I .
Recall that the asymptotic errors for the one-dimensional
problem
(3.13) are 0(exp(-rcd/ht ) ) , 0(exp(-YMth t.) ) , and
0(exp(-6Ntht ))
(see (3.24)).
bound
which
(3.23)
is
These
depend
analogous to (5.5).
on
the growth
Similarly the
asymptotic errors associated with the spatial problem (3.14)
are 0 (e x p (-ird/hx ) ) , 0(exp(-aMx hx ) ) , and 0 (e x p (-|iNxh x ) .
Mx is chosen,
balancing the asymptotic
dimensional
problems
with
respect
errors for
to
Once
the one­
0(exp(-aMx hx ) )
determines the following stepsizes and summation limits:
(5.8)
hx
=
tc/
(2 a M x )^
where the angle d in Figure 6 is taken to be
(5.9)
Nx =
(5.10)
h t = hx
n / 2 ,
^ Mx + I
and
(5.11)
Mt =
^ Mx + I
Assuming the solution.of
a smaller value of N t ,
(5.12)
Nt =
(3.1) decays exponentially in time,
76
may be chosen (see Lund [6]).
Note that when — M„ or — M
p x
Y x
is an integer these are the values used for Nx and
respectively.
,
The additional +.1 appearing in (5.9) , (5.11),
and (5.12) guarantees that all the appropriate errors are at
least 0(e x p (-aMx hx )).
Parameter selections for the approximate in two or three
space
dimensions
balancing
are
deduced
asymptotic
(3.60)
errors.
via
the
Hence,
u(x,y,t)
of
(5.13)
Iu(x ,y ,t ) I < K x a+^ (I-X)P+^ y^+^
same
if
notion
the
of
solution
satisfies
(l-y)f?+^ tY+^ e-61:
then the additional stepsize by and summation limits
Ny for the approximate
M y and
(3.62) are given by
(5.14)
hy = hx
(5.15)
My
a
Mx + I
7 *
Ny =
^V+1
and
(5.16)
Similarly
for
(3.61),
assuming
the
solution
u (x ,y ,z ,t )
satisfies
(5.17)
|u(x,y,z,t)I
< Kw(x,y,t)zUt%
(i-z)v+^
where
(5.18)
then
w(x,y, t ) = x a+^ (I-X)P+^ y ^ +^
the
approximate
remaining
(3.63)
are
parameters
for
(1-y) rt^
the
tY+^ e~6t
,
Sinc-Galerkin
I
77
(5.19)
hz = hx
(5.20)
Mz =
and
(5.21)
For
a
Mx + I
V
Nz =
all
examples
in
one
sequence of runs with Mx ='4,
or
two
space dimensions a
8, 16, and 32 is reported.
In
three space dimensions a run corresponding to Mx = 32 is not
currently possible,
considerable
unknowns.
machine
to the left.
the
storage
With respect
error result
large to
even on
to all
CRAY
XMP/48,
needed
for
due
the
nine examples,
to the
array
of
whenever an
is reported from a CRAY XMP/48 run a * appears
This *
perform on
also indicates
that the
run was too
the Honeywell machine available to the
author.
In all cases
yields a
reported,
M^
>
Nt
(see
(5.12)), which
much smaller discrete system than the choice of N t
given by (3.26).
The choice M t = Nt or that given by
(3.26)
results in larger matrices with h o corresponding increase in
accuracy.
Worth noting is that as a consequence of
T(t) = An(t),
large.
the map
the sine gridpoints in time become quite
Recalling (3.9),
t^
s e^^t
_
In @11 Q f th^
following results the selection Mx = 16, for example,
to a choice of Nt that yields t ^
= e 3 ^*7 8 5 4 ^ = 1 0 . 5 5 .
leads
A
78
large
number
of
iterations
is
typically
approximate the solution for a t
of similar
using
Hence,
time-marching
schemes.
required
to
magnitude when
in comparison,
the
Sinc-Galerkin method requires much smaller systems to attain
the same
accuracy as finite differences
(see the discussion
following Examples 5.7 - 5.9).
The parameter a = % is common to all nine examples hence
the stepsize h = h x = it/ (2aMx )^ and the asymptotic error
0(exp(-aMx hx )) are the same throughout this chapter.
result;
these
values
are
not
included
Computationally the asymptotic rate
0(exp[-aMx hx ] ) = 0(exp[-%/(2(xMx )%])
in
every t a b l e .
(which for
d =
Exponentially Damped Sine Waves
Damped sine wave in one-dimension. .
L v (x ,t ) 2 v t t (x,t) - v x x (x,t)
= {Jt2 + (2 - 4t + (I + jc2 )t2 )e- t }sin(rcx)
v(0,t)
= v(l,t)
v(x, 0) = Sin(Ttx)
= 0
, v t (x, 0) = 0
This problem is transformed to the form (3.1) via
u(x,t)
= v(x,t)
Lu(x,t)
u(0,t)
- sin(Ttx) , which yields
= (2 - 4t + (I + Tt2 Jt2 Je-t Sin(Itx)
= u(l,t)
= 0
u ( x , 0) = u t (x,0) = 0
The analytic solution is u(x,t)
jr/2 is
is consistently attained
at the sine gridpoints.
Example 5.1:
As a
= t2e- t sin (itx) .
The
79
(5. 4)) are a = P =
Table 2 displays the
for
gridpoints
maximum
the
1/2, Y = 3/2, and 6 = 1 .
absolute
sequence
error
Il
(see
X=
parameters
8,
at
16,
the sine
and
Additionally a column with asymptotic error is given.
a =
|$, M x
= N jf.
In
this case
referred to as a centered sum.
the sum
The sum on
,Since
on i in (5.7)
A in
noncentered sum, as commonly happens when using
Table 2.
3 2.
(5.7)
is
is a
(5.12).
Numerical Results for Example 5. I .
h
HE^1 ) (h) II
Asymptotic Error
I
1.57080
.378-1
.432-1
3
2
I .11072
.253-1
.118-1
16
6
3
.78540
.905-3
.188-2
32
11
4
.55536
.320-3
.138-3
MX
NX
Mt
Nt
4
4
2
8
8
16
32
Example 5.2:
Damped sine wave in two-dimensions.
L u (x ,y ,t ) = (2 - 4t + (I + 2rc2 )t2 )e-tsin(rcx)sin(rcy)
uI
=0
*3(0,I)2
u|
= ut
It=O
,
t=0
where, recall,
cube.
= 0
(0,1)n refers to the open n-dimensional unit
The analytic solution is
u ( x , y ,t ) = t 2e -tsin( jcx)sin( rcy) , hence the parameters are
Q = P = ?=
?? = 1/2, v = 3/2, and 6 = 1.
Table 3 lists
IIE^2 ) (h) II, the maximum absolute error at the sine
80
gridpoints.
Note that
the result corresponding.to M„ = 32
was obtained on the CRAY XMP/48 as indicated by the *.
Table 3.
Numerical Results for Example 5.2
IIE^2 ) (h) Il
MX
NX
MY
NY
Mt
Nt
4
4
4
4 .
2
I
.813-1
8
8
8
8
3
2
.760-1
16
16
16 :
16
6
3
.387-3
32
32
32
32
11
4
*.451-3
Damped sine wave in three- d i m e n s i o n s .
Example 5.3:
L u (x ,y ,z ,t) = (2 - 4t + (I + 3n 2 )t 2 )e -tsin( rcx) sin( jry)sin( jcz ;
= 0
3 (0 ,1 )^
t=0
\
= 0
= u,
't=0
The true solution u(x,y;z,t)
the
three-dimensional
solutions.
4.
= t2e-tsin(
analogue
of
jtx )sin
the
The results for this problem are
(iry)sin( Jtz) is
previous
two
given in Table
81
Table 4.
mX
Numerical Results for Example 5.3.
nX
4
4
My Hy
Mz V
Mt
Mt
HEj (H)II
4
4
2
I
.325-0
4
4
13 1
8
8
8
8
8
8
3
2
* .123-1
16
16
16
16
16
16
6
3
* .451-3
With respect
to Examples 5.1, 5.2, and 5.3,
absolute value of the true solution
indicates
that
there
is
no
is the
the maximum
same.
Table 5
consistent difference in the
error results for the various space dimensions.
Table 5.
Mx
Numerical Results for the Damped Sine Wave.
h
IIE^lj (h)ll
IIE^) (h)||
IIE^3 * (h)||
Asymptotic
Error
4
1.57080
.378-1
.813-1
.325-0
.432-1
8
I .11072
.253-1
.760-1
*.123-1
.118-1
16
.78540
.905-3
.387-3
*.451-3
.188-2
32
.55536
.320-3
*.451-3
wmm —
mtm — ■
.138-3
82
Singular Problems
Example 5.4:
A singular problem in one space dimension.
(3 - 12t + 4t2 ) x An(x)
L u (x ,t )
u(0,t)
= u(l,t)
ITt
= 0
u(x,0) = U t ( X fO) = 0
The true solution for this problem is
u(x , t )
=
t 3/ 2e -t
singular at t =
Although
ux
satisfied.
x
0 and
is
the
again h
at
x
=0,
method
requires
no
at x
= 0.
condition (3.36)
despite the different
regard to implementation.
stepsize is
The solution is algebraically
logarithmically singular
unbounded
Further,
singularities
An(x).
is
character of the
modification
For a = p = % and Y = 6
=1,
with
the
= hx = ht = k / (Mx )^ and the asymptotic
rate shown in Table 6 is
achieved,
despite
the presence of
singularities.
Numerical Results for Example 5.4.
Table 6.
Mt
Nt
IIEy ) (h) Il
Asymptotic Error
MX
NX
4
4
.2
I
.321-2
.432-1
8
8
4
2
.405-2
.118-1
16
16
8
3
.852-3
.188-2
32
32
16
4
.767-4
.138-3
83
In fact,
comparing Tables 6 and 2, this problem has slightly
smaller errors
with
a
minor
increase
in
the
number of
gridpoints near t = 0 due to the algebraic singularity.
logarithmic singularity
affects neither
The
the performance of
the method nor the system size.
Example 5.5:
A singular problem in two space dimensions.
(3 - 12t + 4t2 ) x An(x)y An(y)
L u (x ,y ,t )
- 4t 2 ' ^ An(y)
uI
+ — An(x)
e
4>TF
=0
'3(0,I ) 2
uI
= utI
It=0
= 0
't=0
The true solution for this problem,
u (x ,t ) = t 3/2e -t x An(x)y A n ( y ) , exhibits the same algebraic
singularity at t = 0 as the previous e x a m p l e .
the
solution
anything,
Example
is
singular
on
the
spatial
In addition,
boundary.
If
the difficulties appear
more severe
here than in
‘
5.4;
h o w e v e r , the
errors
shown
in Table 7 are
slightly better than those, in Table 6. . The parameters used
are ot = fi = C = f? = % and y = 6 = 1 .
84
Table 7.
Numerical Results for Example 5.5.
. '•
IIE^12 > (h)||
MX
Nx
mY
”y
Mt
Nt .
4
4
4
4
2
I
.221-2
8
. 8
8
8
4
2
.370-2
16
16
16
16
8
3
.235-3
32
32
32
32
16
4
*.530-4
A singular problem in three space d i m e n s i o n s .
Example 5.6:
(3 - 12t + 4 t 2 ) x An(x)y An(y)z An (z )
Lu (x ,y ,z ,t ) =
- 4t 2
+ —
z
uj
^
An(y)An(z)
An(x)An(y)
+ ^
An(x)An(z)
e
U t
=0
'3(0,I)3
U j
= UtI
*t=0
= 0
't=0
The true solution is
u(x,y,z, t)
=
t 3/,2e -t
x
An(x)y An(y)z An(z) and, with the
addition of u = v = ?£, the parameters are identical to those
in Example
5.5.
Table 8 lists the results for the present
example while Table 9
singular p r o b l e m s .
includes the
errors for
each of the
85/
Table 8.
Numerical Results for Example 5.6.
IIEjl3 ) (h)||
MX
Nx
mY
NY
Mz
Nz
Mt
Nt
4
4
4
4
4
4
2
I
.354-2
8
8
8
8
8
8
4
2
*.286-3
16
16
16
16
16
16
8
3
*.405-4
Table 9.
MX
Numerical Results for the Singular P r o b l e m s .
h
I l ) (h)ll
IIEjl2 ) (h) II
IIEjl3 ) (h) II
Asymptotic
Error
4
1.57080
.321-2
.221-2
.354-2
.432-1
8
I .11072
.405-2
.370-2
*.286-3
.118-1
16
.78540
.852-3
.235-3
*.405-4
.188-2
32
.55536
.767-4
*.530-4
.138-3
—
For runs corresponding to Mx = 8 and 16 the error associated
with Example 5.6 is almost a full decimal
the
singular
Moreover,
this
singularity of
examples
in
improvement
the problem.
using finite differences.
one
and
occurs
. No
place better than
two space dimensions.
despite
the
greater
such analogue exists when
86
Problems Dictating Noncentered Sums in All Indices
Example 5.7:
Lu(x,t)
Noncentered sums in one space dimension.
= {(2 - 4t + t2 ) x (I - x ) 3 - 6t2 (l - x)(2x - I ) }e_t
u(0,t)
= u(l,t)
= O
U(XfO)
= u t (x,0) = 0
The analytic solution of this problem is
u(x,t)
=
t 2e -tx(l - x ) 3 .
The parameter selections a = 1/2,
P = 5/2, Y = 3/2, and 6 = 1
space
and
time
errors in Table 10.
Table 10.
f
(Mx
Nx
dictate noncentered sums in both
and M t ^ Nt ) ^as shown with the
Again h s hx = h t = %/(Mx )%.
Numerical Results for Example 5.7.
IIE^l) (h)||
Asymptotic Error
Mt
Nt
I
2
I
.237-2
.432-1
8
2
3
2
.208-2
.118-1
16
4
6
.707-4
.188-2
32
7
11
.237-4
.138-3
MX
NX
4
.
3
4
Even though there are three-fourths fewer
the region
gridpoints in
% < x < I , the same asymptotic rate e x p (-Mxh x / 2)
is predicted.
In
comparison to
Examples 5.1
and 5.4 the
results are better even with the smaller system size.
This
87
is because
the convergence rate of the Sinc-Galerkin method
is governed by the asymptotic behavior of the solution.
Example 5.8:
Noncentered sums in two space dimensions.
Lu(x,y,t)
= 10{(2 - 4t + t 2 ) x
(I - x ) 3 y 2 (l - y)
- 2t2 [3(I - x)(2x - l)y2 (l - y )
+ x ( I - x) 3 (l - 3y)]}e-t
uI
=o
'3(0,I)2
= UtI
t=0
=0
U=O
The solution of this problem,
u(x,y,t)
= 10t 2e~tx(i - x ) 3 y 2 (l - y ) ,
selections, a
6=1.
= 1/2,
|3 = 5/2, £ = 3/2, v = 1/2, y = 3/2, and
As a result of
appearing in
these parameter
the approximate
more complete discussion of
comparison to
example.
appearing on
selections the sums
(3.62) are all noncentered.
how
small
this
system
A
is in
a finite difference solution follows the next
Note
that
the
multiplicative
factor
of
10
the right - hand side of the problem was chosen
so that the solution was of the same
the previous
yields the parameter
example.
current e x a m p l e .
Table 11
order of
magnitude as
gives the results for the
88
Table 11.
Numerical Results for Example 5.8.
Mt
Nt
4
2
I
.805-2
3
8
3
2
.489-2
4
6
16
6
3
.867-4
7
11 .
32
11
4
*.682-3
MX
Nx
“y
V
4
I
2
8
2
16
32
. Example 5.9:
IIE^2 ) (h) Il
Noncentered sums in three space dimensions.
L u (x ,y ,z ,t ) = 100{(2 - 4t + t 2 )x(I - x ) 3Y 2 (I - y ) z 3 (I - z ) 2
- 2t2 [3(I - x)(2x - I )y 2 (I - y ) z 3 (I - z ) 2
+ x ( I - x)3 (I - 3 y )Z3 (I - z)2
+ x ( I - x)3y 2 (I - y ) (IOz3 - 12z2 + 3 z ) ] } e ~ ^
u|
=0
I3(0,1)3
u|
= ut|
It=O
= 0
.
*t=0
The analytic solution of this problem is
u(x,y,z, t)
=
IOO t 2S."tXt I
-
x)3
y 2 (I
-
y ) z 3 (I
Analogous to the factor of 10 in Example 5.8,
100 is
a magnitude a d j u s t m e n t .
P = 5/2,
6 = 1
the
z) 2 .
factor of
The parameters a = 1/2,
f = 3/2, 7) = 1/2 , u - 5/2,
v = 3/2,
Y = 3/2, and
yield the summation parameters appearing
in Table 12.
The composite error results for Examples 5.7 - 5.9 are given
in Table 13.
89
Table 12.
■
Numerical Results for Example 5.9.
MX
Nx
My
NY
4
I
2
8
2
16
4
Table 13.
MX
HEjl3 > (h) Il
Mz
Nz
Mt
Nt
4
I
2
2
I
.395-2
3
8
2
3
3
2
*.389-3
6
16
4
6
6
3
*.313-4
.
Numerical Results for Problems Dictating
Noncentered Sums in All Indices.
I
h
l
(h) Il
IlE u 2 ^ (%) Il
IlE^S ) (h) Il
Asymptotic
Error
4
1.57080
.237-2
.805-2
.395-2
.432-1
8
1.11072
.208-2
.489-2
*.389-3
.118-1
16
.78540
.707-4
.867-4
*.313-4
.188-2
32
.55536
.237-4
*.682-3
The
last
three
examples
_____
illustrate how the parameter
selections described by (5.8) - (5.12),
(5.19)
-
effort for
the
(5.21)
(5.14) - (5.16),
and
minimize the expenditure of computational
the Sinc-Galerkin
method's
.138-3
competitiveness
like finite differences.
As
method.
This also increases
with respect to alternatives
discussed
in
Chapter
4, one
measure of competitiveness is the number of nonzero elements
in the coefficient matrices.
I
For instance,
the asymptotic
90
error
corresponding
to
Maintaining
this
requires
stepsize
domain
a
to
at
gridpoints on
time.
error
least
h FD
t
that
matrices
This evidence
the
=
=
using
=
16
finite
is .188-2.
differences
.04 and iteration of the time
6.3.
This
translates
scheme as
674,200 nonzero
5.8 and 21,595,000 for
entries.
when
M jj
finite difference
matrix with
Sinc-Galerkin
using
computational
Example
have
author
herein are,
notes
one system
elements for Example
5.9.
Their corresponding
251,160 and 3,294,060 nonzero
supports
S t e n g e r 's
efficiency
of
the
[13] statement
Sinc-Galerkin
method becomes more apparent in higher dimensions.
the
to 25
any of the spatial intervals and 158 steps in
Solving the
yields a
runs
Further,
that the computational savings exhibited
in large p a r t , due to including
in the Galerkin p r o c e d u r e .
the time domain
REFERENCES CITED
Farlow, S.J.
Partial Differential Equations for
Scientists and E n g i n e e r s , John Wiley and Sons,
New York, 1982.
^
Weinberger, H .F . A First Course in Partial
Differential E q u a t i o n s , John Wilqy and Sons,
New York, 1965.
Botha, J .F . and Finder, G .F . Fundamental Concepts in
the Numerical Solution of Differential E q u a t i o n s ,
John Wiley and Sons, 1983.
Ames, W.F.
Equations,
Numerical Methods for Partial Differential
2nd e d . , Academic Press, New York, 1977.
S t e n g e r , F . "A Sinc-Galerkin Method of Solution of
Boundary Value Problems."
Mathematics of Computation
33 (January 1979):
85-109.
Lund, J.
"Symmetrization of the Sinc-Galerkin Method
for Boundary Value P r o b l e m s ." Mathematics of
Computation 47 (October 1986):
571-588.
Gottlieb, D . and O r s z a g , S.A.
Numerical Analysis of
Spectral Methods:
Theory and A p p l i c a t i o n , SIAM,
Philadelphia, 1977.
Whittaker, E .T . "On the Functions Which Are
Represented by the Expansions of the Interpolation
T h e o r y . " P r o c . Roy. S o c . Edinburgh 35 (1915):
181-194.
Whittaker, J.M.
Interpolatory Function T h e o r y ,
Cambridge University P r e s s , London, 1935.
M c N a m e e , J., S t e n g e r , F., and Whitney, J.L.
"Whittaker's Cardinal Function in Retrospect."
Mathematics of Computation 25 (January 1971):
141-154.
S t e n g e r , F . "The Approximate Solution of ConvolutionType Integral Equations."
SIAM J . Math. Anal. 4
(August 1973):
536-555.
92
REFERENCES CITED— Continued
12.
S t e n g e r , F . "Integration Formulas via the
Trapezoidal Rule."
J . Inst. Maths. Applies.
(1973):
103-114.
12
13.
S t e n g e r , F.
"Numerical Methods Based on Whittaker
Cardinal, or Sine Functions."
SIAM Review 23 (April
1981) : 165-223.
14.
G e a r , W .C . Numerical Initial Value Problems in
Ordinary Differential E q u a t i o n s , Prentice-Hall,
Englewood Cliffs, New Jersey, 1971.
15.
McArthur, K . M . , B o w e r s , K.L., and Lund, J.
"Numerical
Implementation of the Sinc-Galerkin Method for SecondOrder Hyperbolic Equations."
To appear in Numerical
Methods for Partial Differential E q u a t i o n s , (1987).
16.
Davis, P.J.
Circulant M a t r i c e s , John Wiley and Sons,
New York, 1979.
17.
Isaacson, E . and Keller, H .B . Analysis of Numerical
Methods, John Wiley and Sons, New York, 1966.
18.
d e B o o r , C. and S w a r t z , B . "Collocation Approximation
to Eigenvalues of an Ordinary Differential Equation:
Numerical Illustrations." Mathematics of Computation
36 (January 1981):
1-19.
■.
'
MONTANA STATE UNIVERSITY LIBRARIES
762 1002 490 5