1. Programming assignment # 3 Numerical Methods for PDEs Spring 2007
Jim E. Jones

2. Programming Assignment #3
Problem 1
Problem 2
K=.01
K=1
K=100
K=1

3. Programming Assignment #3
Use your code to investigate these issues
How does the variable K effect the computed solution. This is best seen by comparing plots of the solution to the two problems.
How does the number of conjugate gradient iterations required compare to the number of Gauss-Seidel iterations required for the same tolerance.

4. Programming Assignment #3
On web page are three matlab codes:
hw3gs: Gauss-Seidel
hw3cg: Conjugate Gradient
hw3pcg: Preconditioned Conjugate Gradient

5. Plots showing the effect of variable K on the solution
Constant K
Variable K

6. Results: Iterations needed for tol=1e10-6
K=1
n=10
n=20
n=40
n=80 GS
132
474
1673
5790 CG 23 48 95
185
variable K
n=10
n=20
n=40
n=80 GS
179
662
2424
8786 CG
227
1032
3106
7676

7. Conjugate gradient algorithm
To solve Au=f, starting from initial guess u0
r0=f-Au0
for k=1,2,3,…
rk-1= (rk-1Trk-1)
if k=1
p1= r0
else
bk-1=rk-1/rk-2
pk=rk-1+bk-1pk-1
end if
qk=Apk
ak= rk-1/(pkTqk)
xk=xk-1+akpk
rk=rk-1-akqk
Check convergence
end for

8. Preconditioning
The convergence of conjugate gradients for solving Ax=b depends on the eigenvalues of A (see Shewchuck).
Sometimes it pays to modify the system by multiplying by a matrix, M-1Ax=M-1b. Now the convergence depends on eigenvalues of M-1A.

9. Preconditioned conjugate gradient algorithm
To solve Au=f, starting from initial guess u0
r0=f-Au0
for k=1,2,3,…
sk-1= M-1rk-1
rk-1= (rk-1Tsk-1)
if k=1
p1= s0
else
bk-1=rk-1/rk-2
pk=sk-1+bk-1pk-1
end if
qk=Apk
ak= rk-1/(pkTqk)
xk=xk-1+akpk
rk=rk-1-akqk
Check convergence
end for

10. Preconditioning The preconditioner M should be “close” to A.
The preconditioner should be easy to invert.
Simple preconditioner, M=D, the diagonal of A.

11. Results: Iterations needed for tol=1e10-6
K=1
n=10
n=20
n=40
n=80 GS
132
474
1673
5790 CG 23 48 95
185
PCG
variable K
n=10
n=20
n=40
n=80 GS
179
662
2424
8786 CG
227
1032
3106
7676
PCG 32 68
133
255

12. Upcoming Schedule March M W 12 14 19 21 26 28 April M W 2 4 9 11 16 18
April
M W
Upcoming Schedule
Take home portion of exam handed out March 28
Take home due and in class exam April 2
Programming assignment #4 due April 9
Final Project presentations April 23 & 25

13. Upcoming Schedule March M W 12 14 19 21 26 28 April M W 2 4 9 11 16 18
April
M W
Upcoming Schedule
Take home portion of exam handed out March 28
Take home due and in class exam April 2
Programming assignment #4 due April 9
Final Project presentations April 23 & 25
Optional: Will drop lowest programming assignment 13

14. Optional Programming assignment #4
Implement the finite difference method we talked about last time for the hyperbolic PDE:
Exact solution

15. Optional Programming assignment #4
Investigate stability and accuracy issues
What relationship between h and k must hold for stability? Do your results agree with the CFL condition?
How does the error behave:
O(h+k)?
O(h2 + k)?
O(h2 + k2)?
???