Post on 04-Jan-2016
transcript
Programming assignment #2Solving a parabolic PDE using finite differences
Numerical Methods for PDEs Spring 2007
Jim E. Jones
The Problem
lxxfxu
ttlutu
tlxtxx
utx
t
u
0),()0,(
0,0),(),0(
0,0),,(),(2
22
Reference: Burden and Faires, Numerical Analysis, Ch12
The Problem
lxxfxu
ttlutu
tlxtxx
utx
t
u
0),()0,(
0,0),(),0(
0,0),,(),(2
22
Reference: Burden and Faires, Numerical Analysis, Ch12
PDE
Boundary conditions
Initial conditions
Last lecture we will looked at three methods
• Forward Difference method (explicit)
• Backward difference method (implicit)
• Crank-Nicolson method (implicit)
Time stepping formula
Forward Difference method
2
2
h
kr
)(2
1 ,1.122
,2
2
1, jijijiji wwh
kw
h
kw
i
j
h
k
Let
)),21(,(
1
rrrtridiagB
wBw jj
Time stepping formula
Backward Difference Method
)),21(,(
1
rrrtridiagA
wwA jj
1,,12
2
,12
2
,2
2
21
jijijiji ww
h
kw
h
kw
h
k
i
j
h
k
2
2
h
kr
Let
Time stepping formula
Crank-Nicolson Method
022
2 2
1,11,1,1
2
,1,,12
1,,
h
www
h
www
k
ww jijijijijijijiji
i
j
h
k
2),1(,
2
2),1(,
2
1
rr
rtridiagB
rr
rtridiagA
wBwA jj
Time stepping formula
All Methods
111
1
jjjjj wTwBAwwBwA
1 jj wBwA
Time stepping corresponds to matrix multiplication
Approximations are generated by repeated application of T
0wTw kk
Approximations are generated by repeated application of T
If the initial conditions are perturbed, the perturbed approximation is
And their difference is
Stability
0wTw kk
)( 0
wTz kk
k
kk Twz
Following the material from ODEs we will say that a numerical method is stable if perturbations in the initial conditions give rise to perturbations in the computed solution that do not grow without bound.
We want dk to be bounded for all k.
Stability
kkkk Twzd
Following the material from ODEs we will say that a numerical method is stable if perturbations in the initial conditions give rise to perturbations in the computed solution that do not grow without bound.
We want dk to be bounded for all k.
Stability
kkkk Twzd
What properties of the matrix T do we need to guarantee dk doesn’t grow without bound?
Following the material from ODEs we will say that a numerical method is stable if perturbations in the initial conditions give rise to perturbations in the computed solution that do not grow without bound.
We want dk to be bounded for all k.
Stability
kkkk Twzd
We need the right analogue of absolute value which is what we had for the scalar case in the ODE slides.
A vector norm is a function ||x|| from Rn to R with the following properties:
Review: Vector Norms
||||||||||||
||||||||||
00||||
0||||
yxyx
xx
xx
x
Review: Common Vector Norms
n
ii
ini
n
ii
xxl
xxl
xxl
111
1
2/1
1
222
||||:||
||max||:||
||:||
A matrix norm is a function ||A|| from the set of n x n matrices to R with the following properties:
Review: Matrix Norms
||||||||||||
||||||||||||
||||||||||
00||||
0||||
BAAB
BABA
AA
AA
A
Any vector norm ||x|| defines a corresponding matrix norm via:
And for this natural, matrix norm and its corresponding vector norm we have the following inequality
Review: Natural Matrix Norms
||||max||||1||||
xAAx
|||||||||||| xAxA
The infinity norm of a matrix is the maximum row sum of the absolute values of its entries.
The one norm of a matrix is the maximum column sum of the absolute values of its entries.
Review: Common Natural Matrix Norms
n
jij
niaA
11
||max||||
n
iij
njaA
11
1 ||max||||
The spectral radius (A) of a matrix is the largest eigenvalue in absolute value
The two norm of a matrix is related to the spectral radius
and if A is symmetric
Review: Spectral Radius and Two Norm
||max)( xxA
A
2/12 )]([|||| AAA t
)(|||| 2 AA
Following the material from ODEs we will say that a numerical method is stable if perturbations in the initial conditions give rise to perturbations in the computed solution that do not grow without bound.
Stability
||||||||||||||||||||
kk
k
kkkk
TTd
Twzd
We need conditions to guarantee
1|||| T
Stability of Forward Differences using infinity norm.
|21|2||||
)21(00
)21(00
0)21(00
00)21(0
00)21(
00)21(
rrT
rr
rrr
rrr
rrr
rrr
rr
BT
Td kk
2
2
h
kr
Stability of Forward Differences using infinity norm.
|21|2|||| rrT 2
2
h
kr
114)21(2|21|2|||| rrrrrT
• If r > ½
Unstable
•Else
Stable
1)21(2|21|2|||| rrrrT
Stability of Forward Differences using two norm.
)(||||
)21(00
)21(00
0)21(00
00)21(0
00)21(
00)21(
2 AT
rr
rrr
rrr
rrr
rrr
rr
BT
Td kk
2
2
h
kr
Eigenvalues and Eigenvectors on nxn matrix T
)21(00
)21(00
0)21(00
00)21(0
00)21(
00)21(
rr
rrr
rrr
rrr
rrr
rr
BT
1sin)(
,...,2,1,)1(2
sin41 2
n
imv
nmn
mr
im
m
Stability requires spectral radius bounded by one
1|)1(2
sin41|max)( 2
1
n
mrT
nm
• If r > ½Unstable
•ElseStable
Same result as infinity norm
Assignment #2
• Forward Difference method (explicit)
• Backward difference method (implicit)
• Crank-Nicolson method (implicit)
lxxfxu
ttlutu
tlxtxx
utx
t
u
0),()0,(
0,0),(),0(
0,0),,(),(2
22
Assignment #2 will is due Wednesday Feb 21. You will code up these three methods for a particular problem and look at accuracy and stability issues.
PDE solution is:
The Burden and Faires text contains results for this example
Assignment #2
lxxfxu
ttlutu
tlxtxx
utx
t
u
0),()0,(
0,0),(),0(
10,0),,(),(2
22
)sin()(,1,1 xxfl
)sin(),(2
xetxu t
Your job is to experiment with different values of h and k. Do your best to investigate numerically some of the issues we’ve talked about in the lecture.
• Stability: Run at least two problems with forward differences. One that satisfies the stability condition and one that does not. Comment on your observations. We’ve not seen it yet, but the other two methods are unconditionally stable.
•Convergence: Backward and Forward differencing has truncation error O(k+h2). Crank-Nicolson is O(k2+h2). Calculate the errors you see and comment on how they agree, or not, with these truncation error results.
•Comparison: Comment on the relative strengths and weaknesses of the three methods.
Assignment #2