16.920J/SMA 5212
Numerical Methods for Partial Differential Equations
Lecture 5
Finite Differences: Parabolic Problems
B. C. Khoo
Thanks to Franklin Tan
19 February 2003
16.920J/SMA 5212 Numerical Methods for PDEs
OUTLINE
•
•
•
•
•
•
Governing Equation
Stability Analysis
3 Examples
Relationship between σ and λh
Implicit Time-Marching Scheme
Summary
Slide 2
GOVERNING EQUATION
Consider the Parabolic PDE in 1-D
∂u
∂ 2u
=υ 2
∂t
∂x
x ∈ [ 0, π ]
subject to u = u0 at x = 0, u = uπ at x = π
uπ
u0
u ( x, t ) = ?
x =π
x=0
If υ ≡ viscosity → Diffusion Equation
If υ ≡ thermal conductivity → Heat Conduction Equation
Slide 3
STABILITY ANALYSIS
Discretization
Keeping time continuous, we carry out a spatial
discretization of the RHS of
∂u
∂ 2u
=υ 2
∂t
∂x
x =π
x=0
x0
x1
xN −1 xN
x2
2
16.920J/SMA 5212 Numerical Methods for PDEs
There is a total of N + 1 grid points such that x j = j ∆x,
j = 0,1, 2,...., N
Slide 4
STABILITY ANALYSIS
Discretization
∂ 2u
∂x 2
Use the Central Difference Scheme for
∂ 2u
∂x 2
=
j
u j +1 − 2u j + u j −1
∆x 2
+ O ( ∆x 2 )
which is second-order accurate.
•
Schemes of other orders of accuracy may be constructed.
Slide 5
Construction of Spatial Difference Scheme of Any Order p
The idea of constructing a spatial difference operator is to represent the spatial
differential operator at a location by the neighboring nodal points, each with its own
weightage.
The order of accuracy, p of a spatial difference scheme is represented as O ( ∆x p ) .
Generally, to represent the spatial operator to a higher order of accuracy, more nodal
points must be used.
j−2
•
j−1
•
du
dx
j
•
j
j+1
•
j+2
•
du
Consider the following procedure of determining the spatial operator
dx
order of accuracy O ( ∆x 2 ) :
3
up to the
j
16.920J/SMA 5212 Numerical Methods for PDEs
1.
du
Let
dx
be represented by u at the nodes j−1, j, and j+1 with α −1 , α 0 and
j
α1 being the coefficients to be determined, i.e.
2.
du
dx
+ α −1u j −1 + α 0 u j + α1u j +1 = O ( ∆x p )
(1)
j
Seek Taylor Expansions for u j −1 , u j and u j +1 about u j and present them in a
table as shown below.
(Note that p is not known a priori but is determined at the end of the analysis
when the α’s are made known.)
This column consists of all the terms on the
LHS of (1).
uj
uj′
u j′ ′
uj′ ′′
u j′
0
1
0
0
α −1u j −1
α −1
−∆x ⋅α −1
1 2
∆ x ⋅ α −1
2
1
− ∆ x 3 ⋅ α −1
6
α 0u j
α0
0
0
0
α1u j +1
α1
∆x ⋅α1
1 2
∆x ⋅ α 1
2
1 3
∆x ⋅ α 1
6
S1
S2
S3
S4
u j′ +
k =1
k =−1
αk u j+k
Each cell in this row comprises the sum
of its corresponding column.
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16.920J/SMA 5212 Numerical Methods for PDEs
where S1 = (α −1 + α 0 + α1 ) u j
S2 = (1 − ∆x ⋅α −1 + ∆x ⋅ α1 ) u j′
1
1
S3 = ∆x ⋅ α −1 + ∆x 2 ⋅α1 u j′′
2
2
2
1
1
S4 = − ∆x3 ⋅α −1 + ∆x3 ⋅ α1 u j′′′
6
6
k =1
∴ u j′ +
3.
k =−1
α k u j + k = S1 + S 2 + S3 + S 4 + ....
Make as many Si ’s as possible vanish by choosing appropriate α k ’s.
In this instance, since we have three unknowns α −1 , α 0 and α1 , we can
therefore set:
S1 = 0
S2 = 0
S3 = 0
(Note that in the Taylor Series expansion, one starts off with the lower-order
terms and progressively obtain the higher-order terms. We have deliberately
set the Si pertaining to the lower-order terms to zero, thereafter followed by
increasingly higher-order terms.)
Hence,
0 1 1 α −1
1 −1 0 1 α 0 = −
∆x 1 0 1 α1
1
0
Solving the system of equations, we obtain
1
2 ∆x
α0 = 0
α −1 =
α1 = −
1
2 ∆x
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4.
Substituting the α k ’s into u j′ +
u j′ −
k =1
k =−1
α k u j + k = S1 + S 2 + S3 + S 4 + .... yields
1
1
u j +1 − u j −1 ) = ∆x 2 ⋅ u j′′′ + higher-order terms
(
2 ∆x
6
In other words,
du
u j′ =
dx
=
u j +1 − u j −1
+ O ( ∆x 2 ) + ....
2 ∆x
j
O ( ∆x p ) , p = 2
i.e. the above representation is accurate up to O ( ∆x 2 ) .
Some useful points to note:
1.
These 4 steps are the general procedure used to obtain the representation of the
spatial operator up to the order of accuracy O ( ∆x p ) .
2.
d 2u
For other spatial operators, say
dx 2
du
, we simply replace
dx
j
in (1) with
j
the said spatial operator.
3.
For one-sided representations, one can choose nodal points u j + k , k ≥ 0 . This
may be important especially for representations on a boundary. For example
du
dx
+ α 0u j + α1u j +1 + α 2u j + 2 + .... = O ( ∆x p )
j
One possibility is
du
dx
+
j
3u j − 4u j +1 + u j + 2
2 ∆x
= O ( ∆x 2 )
which is also second-order accurate.
(We can also use a similar procedure to construct the finite difference scheme
of Hermitian type for a spatial operator. This is not covered here).
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16.920J/SMA 5212 Numerical Methods for PDEs
STABILITY ANALYSIS
Discretization
du1
υ
= 2 (uo − 2u1 + u2 )
dt ∆x
du2
υ
x2 :
= 2 (u1 − 2u2 + u3 )
dt
∆x
We obtain at x1 :
xj :
du j
dt
xN −1 :
=
υ
∆x 2
(u j −1 − 2u j + u j +1 )
du N −1
υ
= 2 (u N − 2 − 2u N −1 + u N )
∆x
dt
Note that we need not evaluate u at x = x0 and x = xN
since u0 and uN are given as boundary conditions.
Slide 6
STABILITY ANALYSIS
Matrix Formulation
Assembling the system of equations, we obtain
du1 −2
dt
du2
dt
1
=
du j υ ∆x 2 0
1
1
−2
dt
1
du N −1 0
−2
1
1
u1
u2
uj
1 υ uo 2 ∆x 0 0 +
0 υu N −2 u N −1 2
∆x
dt
A
Slide 7
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16.920J/SMA 5212 Numerical Methods for PDEs
STABILITY ANALYSIS
PDE to Coupled ODEs
Or in compact form
du
= Au + b
dt
where u = [ u1
b=
u N −1 ]
T
u2
υ uo
∆x 2
0
0
0
υu N
∆x 2
T
We have reduced the 1-D PDE to a set of Coupled ODEs!
Slide 8
STABILITY ANALYSIS
Eigenvalue and Eigenvector of Matrix A
If A is a nonsingular matrix, as in this case, it is then
possible to find a set of eigenvalues
λ = {λ1 , λ2 ,...., λ j ,...., λN −1}
from det ( A − λ I ) = 0.
For each eigenvalue λ j , we can evaluate the eigenvector V j
consisting of a set of mesh point values vi j , i.e.
T
V j = v1j
v2j
vNj −1
Slide 9
STABILITY ANALYSIS
Eigenvalue and Eigenvector of Matrix A
The ( N − 1) × ( N − 1) matrix E formed by the ( N − 1) columns
V j diagonalizes the matrix A by
E −1 AE = Λ
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16.920J/SMA 5212 Numerical Methods for PDEs
λ1
0
λ2
where Λ =
0
λN −1
Slide 10
STABILITY ANALYSIS
Coupled ODEs to Uncoupled ODEs
du
Starting from
= Au + b
dt
Premultiplication by E −1 yields
E −1
du
= E −1 Au + E −1 b
dt
du
E
= E −1 A ( EE −1 ) u + E −1 b
dt
−1
I
du
E −1
= ( E −1 AE ) E −1 u + E −1 b
dt
Λ
Slide 11
STABILITY ANALYSIS
Coupled ODEs to Uncoupled ODEs
Continuing from
E −1
du
= ΛE −1 u + E −1 b
dt
Let U = E −1u and F = E −1b , we have
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16.920J/SMA 5212 Numerical Methods for PDEs
d
U = ΛU + F
dt
which is a set of Uncoupled ODEs!
Slide 12
STABILITY ANALYSIS
Coupled ODEs to Uncoupled ODEs
Expanding yields
dU1
= λ1U1 + F1
dt
dU 2
= λ2U 2 + F2
dt
dU j
dt
= λ jU j + Fj
dU N −1
= λ N −1U N −1 + FN −1
dt
Since the equations are independent of one another, they
can be solved separately.
The idea then is to solve for U and determine u = EU
Slide 13
STABILITY ANALYSIS
Coupled ODEs to Uncoupled ODEs
Considering the case of b independent of time, for the
general j th equation,
U j = c j eλ jt −
1
λj
Fj
is the solution for j = 1,2,….,N−1.
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16.920J/SMA 5212 Numerical Methods for PDEs
( )
Evaluating, u = EU = E ceλt − EΛ −1E −1 b
Complementary Particular (steady-state)
solution
(transient) solution
( )
where ceλt = c1eλ1t
c2eλ2t
cje
T
λ jt
cN −1eλN −1t The stability analysis of the space discretization, keeping
time continuous, is based on the eigenvalue structure of A.
The exact solution of the system of equations is determined
by the eigenvalues and eigenvectors of A.
Slide 14
STABILITY ANALYSIS
Coupled ODEs to Uncoupled ODEs
We can think of the solution to the semi-discretized problem
( )
u = E ceλt − E Λ −1E −1 b
as a superposition of eigenmodes of the matrix operator A.
Each mode j contributes a (transient) time behaviour of the form
λt
e j to the time-dependent part of the solution.
Since the transient solution must decay with time,
Real ( λ j ) ≤ 0
for all j
This is the criterion for stability of the space discretization (of a
parabolic PDE) keeping time continuous.
Slide 15
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16.920J/SMA 5212 Numerical Methods for PDEs
STABILITY ANALYSIS
Use of Modal (Scalar) Equation
It may be noted that since the solution u is expressed as a
contribution from all the modes of the initial solution,
which have propagated or (and) diffused with the eigenvalue
λ j , and a contribution from the source term b j , all the
properties of the time integration (and their stability
properties) can be analysed separately for each mode with
the scalar equation
dU
= λU + F
dt
j
Slide 16
STABILITY ANALYSIS
Use of Modal (Scalar) Equation
The spatial operator A is replaced by an eigenvalue λ, and
the above modal equation will serve as the basic equation
for analysis of the stability of a time-integration scheme
(yet to be introduced) as a function of the eigenvalues λ
of the space-discretization operators.
This analysis provides a general technique for the
determination of time integration methods which lead to
stable algorithms for a given space discretization.
Slide 17
EXAMPLE 1
Continuous Time Operator
Consider a set of coupled ODEs (2 equations only):
du1
= a11u1 + a12u2
dt
du2
= a21u1 + a22u2
dt
Let u =
u1
u2
, A=
a11
a12
a21
a22
du
= Au
dt
Slide 18
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16.920J/SMA 5212 Numerical Methods for PDEs
EXAMPLE 1
Continuous Time Operator
Proceeding as before, or otherwise (solving the ODEs directly),
we can obtain the solution
u1 = c1ξ11eλ1t + c2ξ12 eλ2t
u2 = c1ξ 21eλ1t + c2ξ 22 eλ2t
ξ11
ξ
and 21 are
ξ 21
ξ 22
where λ1 and λ2 are eigenvalues of A and
eigenvectors pertaining to λ1 and λ2 respectively.
As the transient solution must decay with time, it is imperative that
Real ( λ j ) ≤ 0 for j = 1, 2.
Slide 19
EXAMPLE 1
Discrete Time Operator
Suppose we have somehow discretized the time operator on the
LHS to obtain
u1 = a11u1
n
n −1
+ a12 u2
n −1
+ a22 u2
u2 = a21u1
n
n −1
n −1
where the subscript n stands for the nth time level, then
n
u = Au
n −1
n
where u = u1
Since A is independent of time,
n
u = Au
n −1
= AAu
n−2
n
u2
n T
and A =
= .... = An u
a11
a12
a21 a22
0
In later examples, we shall apply specific time discretization schemes
such as the “leapfrog” and Euler-forward time discretization schemes.
Slide 20
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16.920J/SMA 5212 Numerical Methods for PDEs
EXAMPLE 1
Discrete Time Operator
As
A = E ΛE −1 ,
n
u( = E ΛE −1 ⋅ E ΛE −1 ⋅ .... ⋅ E ΛE −1 ⋅ u&'& ( 0
A
n
A
−1
u = EΛ E u
n
A
where Λ = 0
n
u1 = λ1 ξ11c1 + λ2 ξ12c2
n
n
'
n
'
21 1
n
0
0
λ2n
'
u2 = λ1 ξ c + λ2 ξ 22c2
n
λ1n
where
'
n
c1 ' = E −1 u 0 are constants.
c2 '
Slide 21
Alternative View
Alternatively, one can view the solution as:
U1n
U
n
2
= λ1n
n λ2
U10
U
0
2
where U = E −1u
U n = Λ nU 0
EXAMPLE 1
Comparison
Comparing the solution of the semi-discretized problem where
time is kept continuous
u1
ξ11 ξ12 e λ1t
= [ c1 c2 ] u2
ξ 21 ξ 22 eλ2t
to the solution where time is discretized
!
!
!
u1
ξ11 ξ12 λ1
#
$
% # = [ c1 ' c2 '] $"
% #"
"
u2
ξ 21 ξ 22 $" λ2 n % #
n
n
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16.920J/SMA 5212 Numerical Methods for PDEs
The difference equation where time is continuous has exponential
solution eλt .
The difference equation where time is discretized has power
solution λ n .
Slide 22
EXAMPLE 1
Comparison
In equivalence, the transient solution of the difference
equation must decay with time, i.e.
λn < 1
for this particular form of time discretization.
Slide 23
EXAMPLE 2
Leapfrog Time Discretization
Consider a typical modal equation of the form
du
= λu + aeµt
dt
j
where λ j is the eigenvalue of the associated matrix A.
(For simplicity, we shall henceforth drop the subscript j).
We shall apply the “leapfrog” time discretization scheme given as
du u n +1 − u n −1
=
dt
2h
where h = ∆t
Substituting into the modal equation yields
u n +1 − u n−1
= ( λu + ae µt )
t = nh
2h
n
µ hn
= λ u + ae
Slide 24
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16.920J/SMA 5212 Numerical Methods for PDEs
Reminder
Recall that we are considering a typical modal equation which had been obtained
from the original equation
du
= Au + b
dt
EXAMPLE 2
Leapfrog Time Discretization: Time Shift Operator
u n +1 − u n−1
= λu n + ae µ hn
2h
u n+1 − 2hλu n − u n−1 = 2ha ( e µ hn )
Solution of u consists of the complementary solution c n , and the
particular solution p n , i.e.
un = cn + pn
There are several ways of solving for the complementary and
particular solutions. One way is through use of the shift operator
S and characteristic polynomial.
The time shift operator S operates on c n such that
Sc n = c n +1
S 2 c n = S ( Sc n ) = Sc n +1 = c n + 2
Slide 25
EXAMPLE 2
Leapfrog Time Discretization: Time Shift Operator
The complementary solution c n satisfies the homogenous equation
c n +1 − 2hλ c n − c n −1 = 0
Sc n − 2 hλ c n −
cn
=0
S
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16.920J/SMA 5212 Numerical Methods for PDEs
( S 2 c n − 2hλ Sc n − c n )
( S 2 − 2hλ S − 1)
1
=0
S
cn
=0
S
characteristic polynomial
p ( S ) = (S 2 − 2hλ S − 1) = 0
Slide 26
EXAMPLE 2
Leapfrog Time Discretization: Time Shift Operator
The solution to the characteristic polynomial is
σ1 and σ2 are the two roots
σ (λ h) = S = λ h ± 1 + λ 2 h2
The complementary solution to the modal equation would then be
c n = β1σ 1 + β 2σ 2
n
n
The particular solution to the modal equation is p n =
2aheµ hn e µh
.
e2 µ h − 2hλ e µh − 1
Combining the two components of the solution together,
u n = ( cn ) + ( pn )
(
= β1 λ h + 1 + h 2 λ 2
) + β (λh −
n
2
1 + h 2λ 2
)
n
2ahe µ hn eµ h
+ 2µh
e − 2hλ eµ h − 1
Slide 27
EXAMPLE 2
Leapfrog Time Discretization: Stability Criterion
For the solution to be stable, the transient
(complementary) solution must not be allowed to grow
indefinitely with time, thus implying that
(
= (λh −
)
1+ h λ ) < 1
σ 1 = λ h + 1 + h2 λ 2 < 1
σ2
2
2
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16.920J/SMA 5212 Numerical Methods for PDEs
is the stability criterion for the leapfrog time
discretization scheme used above.
Slide 28
EXAMPLE 2
Leapfrog Time Discretization: Stability Diagram
The stability diagram for the leapfrog (or any general)
time discretization scheme in the σ-plane is
Im(σ )
Region of Stability
Re(σ )
1
-1
Slide 29
Stability Diagram in the λh-plane
Alternatively, we can express the stability criterion for the leapfrog time discretization
scheme as
1
1
λh = σ −
2
σ
s.t. σ < 1
Since σ < 1 and σ = exp(iθ ) ,
Im(λh)
λ h = i sin θ for stability.
The stability diagram for the leapfrog time
discretization scheme in the λh-plane would
therefore be as shown:
1
Re(λh)
-1
18
Region of Stability
16.920J/SMA 5212 Numerical Methods for PDEs
EXAMPLE 2
Leapfrog Time Discretization
In particular, by applying to the 1-D Parabolic PDE
∂u
∂ 2u
=υ 2
∂t
∂x
the central difference scheme for spatial discretization, we obtain
−2 1
1 −2
A=
υ
∆x
0
1
2
0
1
1
−2
which is the tridiagonal matrix
Slide 30
EXAMPLE 2
Leapfrog Time Discretization
According to analysis of a general triadiagonal matrix B(a,b,c), the
eigenvalues of B are
jπ
λ j = b + 2 ac cos
,
N
jπ
λ j = −2 + 2 cos
N
j = 1,..., N − 1
υ
∆x 2
The most “dangerous” mode is that associated with the eigenvalue
of largest magnitude
λmax = −
i.e.
4υ
∆x 2
σ 1 ( λmax h ) = λmax h + λ 2 max h 2 + 1
σ 2 ( λmax h ) = λmax h − λ 2 max h 2 + 1
which can be plotted in the absolute stability diagram.
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16.920J/SMA 5212 Numerical Methods for PDEs
One may note that λ j is always real and negative, thereby satisfying
the criterion for stability of the space discretization of a parabolic
PDE, keeping time continuous.
Slide 31
EXAMPLE 2
Leapfrog Time Discretization: Absolute Stability Diagram for σ
As applied to the 1-D Parabolic PDE, the absolute stability diagram for σ is
Im(σ )
Region of
Instability
Unit circle
σ2 with h
σ1 with h
increasing
increasing
σ1 at h = ∆t = 0
Re(σ )
σ2 at h = ∆t = 0
Region of
Stability
In this case, σ 1 and σ 2 start out being on the unit circle (h = ∆t = 0). However, the
spurious root (refer to following slide) leaves the unit circle as h starts increasing.
Therefore, the spurious root causes the leapfrog time discretization scheme to be
unstable, irrespective of how small h = ∆t is, although it does not affect the accuracy.
The leapfrog time discretization for the 1-D Diffusion Equation is unstable.
Slide 32
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STABILITY ANALYSIS
Some Important Characteristics Deduced
A few features worth considering:
1. Stability analysis of time discretization scheme can be carried out for
all the different modes λ j .
2. If the stability criterion for the time discretization scheme is valid for
all modes, then the overall solution is stable (since it is a linear
combination of all the modes).
3. When there is more than one root σ , then one of them is the principal
root which represents an approximation to the physical behaviour.
The principal root is recognized by the fact that it tends towards one
as λ h → 0, i.e. lim σ ( λ h ) = 1. (The other roots are spurious, which
λ h→0
affect the stability but not the accuracy of the scheme.)
Slide 33
STABILITY ANALYSIS
Some Important Characteristics Deduced
4. By comparing the power series solution of the principal root to eλ h ,
one can determine the order of accuracy of the time discretization
scheme. In this example of leapfrog time discretization,
σ 1 = λ h + (1 + h 2 λ
σ 1 = 1 + hλ +
1
2 2
)
1 1
.−
1 2 2
= λ h + 1 + ( h λ ) + 2 2 .h 4 λ 4
2
2!
h2λ 2
+ ...
2
and compared to
eλ h = 1 + hλ +
h 2λ 2
+ ...
2!
is identical up to the second order of hλ . Hence, the above scheme
is said to be second-order accurate.
Slide 34
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EXAMPLE 3
Euler-Forward Time Discretization: Stability Analysis
Analyze the stability of the explicit Euler-forward time discretization
du u n+1 − u n
=
dt
∆t
as applied to the modal equation
du
= λu
dt
du
where h = ∆t
dt
into the modal equation, we obtain u n +1 − (1 + λ h)u n = 0
Substituting
u n +1 = u n + h
Slide 35
EXAMPLE 3
Euler-Forward Time Discretization: Stability Analysis
Making use of the shift operator S
c n+1 − (1 + λ h)c n = Sc n − (1 + λ h)c n = [ S − (1 + λ h)]c n = 0
characteristic polynomial
Therefore
σ (λ h ) = 1 + λ h
and
c n = βσ n
The Euler-forward time discretization scheme is stable if
σ ≡ 1+ λh < 1
or bounded by λ h = σ − 1
s.t. σ < 1 in the λ h-plane.
Slide 36
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EXAMPLE 3
Euler-Forward Time Discretization: Stability Diagram
The stability diagram for the Euler-forward time
discretization in the λh-plane is
Im(λh)
Unit Circle
Region of Stability
-2
Re(λh)
0
-1
Slide 37
EXAMPLE 3
Euler-Forward Time Discretization: Absolute Stability Diagram
As applied to the 1-D Parabolic PDE, λ = λmax =
−4υ
∆x 2
Im(σ )
σ leaves the unit circle at λh = −2
σ at h = ∆t = 0
-1
σ with h increasing
1
The stability limit for largest h ≡ ∆t =
Re(σ )
−2
λmax
σ leaves the unit circle at σ = −1, i.e. σ = 1+ λh = −1
λ h = −2
h=
−2
since it is the extreme.
λmax
Slide 38
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Predictor-Corrector Time Discretization
Consider the numerical stability of the following predictor-corrector time
discretization scheme
uˆ n +1 = u n + h
u
n +1
du n
dt n
duˆ n +1
1
= u n + uˆ n +1 + h n +1
dt
2
as applied to the typical modal equation
du
= λu + ae µt
dt
of the parabolic PDE. Substituting
du
duˆ
and
into the predictor-corrector scheme
dt
dt
yields
uˆ n +1 = u n + h ( λ u n + ae µ hn )
u n +1 =
where t = n∆t = nh
1 n
u + uˆ n +1 + h ( λuˆ n+1 + ae µ h( n+1) )
2
Utilizing the shift operator
Su n = u n+1
Suˆ n = uˆ n+1
and rearranging the equations into matrix form, we obtain
S
− (1 + λ h )
1
− (1 + λ h ) S
2
1
S−
2
uˆ
n
u
n
h
= 1
ae µ hn
hS
2
To determine the characteristic polynomial, set
Ρ (σ ) = Ρ ( S ) =
S
−
1
(1 + λ h ) S
2
− (1 + λ h )
1 =0
S−
2
24
16.920J/SMA 5212 Numerical Methods for PDEs
1
Ρ (σ ) = Ρ ( S ) = S S − 1 − λ h − λ 2 h 2 = 0
2
σ = 0 (trivial root)
1
2
σ = 1 + λ h + λ 2 h2
i.e. the scheme is a one-root method. Compared to
1
eλ h = 1 + λ h + λ 2 h 2 + ....
2
the scheme is second-order accurate.
To obtain the particular solution, one can perform a matrix inversion and obtain
1
aheµhn (e µh + 1 + λh )
n
2
p =
1
e µh − 1 − λh − λ2 h 2
2
with the complementary solution being
n
1
c = βσ = β 1 + λh + λ2 h 2
2
n
n
The absolute stability diagram (showing λ = −
4υ
) for the 1-D Parabolic PDE is
∆x 2
Im(σ )
Region of
Instability
h increasing further
-1
1
0.5
h increasing from 0
Region of
Stability
25
Re(σ )
16.920J/SMA 5212 Numerical Methods for PDEs
When h increases from zero, σ decreases from 1.0. As h continues to increase, σ
reaches a minimum of 0.5 with λh = −1 and then increases. As h increases further, σ
returns to 1.0 with λh = −2. Prior to this point, the scheme is stable. Increasing h and
thus σ beyond this point renders the scheme unstable.
Hence, this predictor-corrector scheme is stable for small h’s and unstable for large
h’s; the limit for stability is λh = −2 (from above).
In general, we can analyze the absolute stability diagram for the predictor-corrector
time discretization method in terms of
(λ h ) 2
2
σ:
σ (λ h ) = 1 + λ h +
λh :
λ h = −1 ± 2σ − 1
or
λ, the eigenvalue(s) of the A matrix can take on complex forms depending on the
governing equation (as opposed to negative real values for the 1-D parabolic PDE
with central differencing for the spatial derivative).
RELATIONSHIP BETWEEN σ AND λ h
σ = σ(λ
λh)
Thus far, we have obtained the stability criterion of the time
discretization scheme using a typical modal equation. We can
generalize the relationship between σ and λh as follows:
•
Starting from the set of coupled ODEs
du
= Au + b
dt
•
Apply a specific time discretization scheme like the
leapfrog time discretization as in Example 2
du u n +1 − u n−1
=
dt
2h
Slide 39
26
16.920J/SMA 5212 Numerical Methods for PDEs
RELATIONSHIP BETWEEN σ AND λ h
σ = σ(λh)
•
The above set of ODEs becomes
n
u n +1 − u n −1
= Au n + b
2h
•
Introducing the time shift operator S
n
un
Su =
+ 2hAu n + 2hb
S
n
S − S −1
A−
I u n = −b
2h
n
•
Premultiplying E −1 on the LHS and RHS and introducing
I = EE −1 operating on u n
S − S −1
E AE − E
E E −1u = − E −1b n
2h
−1
−1
Λ
Slide 40
RELATIONSHIP BETWEEN σ AND λ h
σ = σ(λ
λh)
•
Putting U n = E −1u n ,
F n = E −1 b
n
S − S −1
we obtain Λ − E
E U n = −F n
2h
−1
S − S −1
2h
S − S −1
i.e. Λ −
U n = −F n
2h
which is a set of uncoupled equations.
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16.920J/SMA 5212 Numerical Methods for PDEs
Hence for each j, j = 1,2,….,N−1,
S − S −1
λj −
U j = − Fj
2h
Slide 41
RELATIONSHIP BETWEEN σ AND λ h
σ = σ(λh)
Note that the analysis performed above is identical
to the analysis carried out using the modal equation
dU
= λU + F
dt
j
All the analysis carried out earlier for a single modal
equation is applicable to the matrix after the
appropriate manipulation to obtain an uncoupled set
of ODEs.
Each j th equation can be solved independently for
U nj and the U nj 's can then be coupled through u n = EU n .
Slide 42
RELATIONSHIP BETWEEN σ AND λ h
σ = σ(λ
λh)
Hence, applying any “consistent” numerical technique
to each equation in the set of coupled linear ODEs is
mathematically equivalent to
1.
Uncoupling the set,
2.
Integrating each equation in the uncoupled set,
3.
Re-coupling the results to form the final solution.
These 3 steps are commonly referred to as the
ISOLATION THEOREM
Slide 43
28
16.920J/SMA 5212 Numerical Methods for PDEs
IMPLICIT TIME-MARCHING SCHEME
Thus far, we have presented examples of explicit time-marching
methods and these may be used to integrate weakly stiff equations.
Implicit methods are usually employed to integrate very stiff
ODEs efficiently. However, use of implicit schemes requires
solution of a set of simultaneous algebraic equations at each
time-step (i.e. matrix inversion), whilst updating the variables at
the same time.
Implicit schemes applied to ODEs that are inherently stable will
be unconditionally stable or A-stable.
Slide 44
IMPLICIT TIME-MARCHING SCHEME
Euler-Backward
Consider the Euler-backward scheme for time discretization
n +1
du
dt
=
u n +1 − u n
h
Applying the above to the modal equation for parabolic PDE
du
= λ u + ae µt
dt
yields
u n +1 − u n
µ n +1 h
= λu n+1 + ae ( )
h
(1 − hλ ) u n+1 − u n = aheµ ( n +1)h
Slide 45
IMPLICIT TIME-MARCHING SCHEME
Euler-Backward
Applying the S operator,
(1 − hλ ) S − 1
u n = ahe
µ ( n +1) h
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16.920J/SMA 5212 Numerical Methods for PDEs
the characteristic polynomial becomes
Ρ (σ ) = Ρ ( S ) = (1 − hλ ) S − 1 = 0
The principal root is therefore
σ=
1
= 1 + λ h + λ 2 h 2 + ....
1 − λh
1
which, upon comparison with eλ h = 1 + λ h + λ 2 h 2 + .... , is only
2
first-order accurate.
The solution is
n
1
U =β
1− λh
n
+
ahe ( )
(1 − λ h ) e µ h − 1
µ u +1 h
Slide 46
IMPLICIT TIME-MARCHING SCHEME
Euler-Backward
For the Parabolic PDE, λ is always real and < 0.
Therefore, the transient component will always tend
towards zero for large n irregardless of h (≡ ∆t).
The time-marching scheme is always numerically stable.
In this way, the implicit Euler/Euler-backward time
discretization scheme will allow us to resolve different
time-scaled events with the use of different time-step
sizes. A small time-step size is used for the short timescaled events, and then a large time-step size used for
the longer time-scaled events. There is no constraint on
hmax.
Slide 47
IMPLICIT TIME-MARCHING SCHEME
Euler-Backward
However, numerical solution of u requires the solution
of a set of simultaneous algebraic equations or matrix
inversion, which is computationally much more
30
16.920J/SMA 5212 Numerical Methods for PDEs
intensive/expensive compared to the multiplication/
addition operations of explicit schemes.
Slide 48
SUMMARY
•
Stability Analysis of Parabolic PDE
Uncoupling the set.
Integrating each equation in the uncoupled set →
modal equation.
Re-coupling the results to form final solution.
•
Use of modal equation to analyze the stability |σ(λh)| < 1.
•
Explicit time discretization versus Implicit time discretization.
Slide 49
Reference:
Numerical Computation of Internal and External Flows, Vol I & II by C. Hirsch,
1992, Wiley Series.
31