CFD Online Logo CFD Online URL
Home > Forums > Main CFD Forum

How to calculate the convergence rate of Gauss-Seidel line iteration?

Register Blogs Members List Search Today's Posts Mark Forums Read

LinkBack Thread Tools Display Modes
Old   November 2, 2009, 23:08
Default How to calculate the convergence rate of Gauss-Seidel line iteration?
Join Date: Mar 2009
Posts: 40
Rep Power: 10
hadesmajesty is on a distinguished road
For the two dimensional Poissonís equation,
second-order centered difference with uniform grid is used to discretize. Then, we can obtain
a[v(n,i+1,j)-2v(n+1,i,j)+ v(n+1,i-1,j)]+c[v(n+1,i,j+1)-2v(n+1,i,j)+ v(n+1,i,j-1)]=0
where a<<c (e.g. a=0.01, c=1), hence y-line iteration should be used.
By using the Fourier expansion of the above equation, we can obtain the convergence rate mu(alpha,beta) as
mu(alpha,beta)= | a/[ 2(a+c)-c*cos(beta)-exp(I*alpha) ] | .
According to some book, the maximal value of the above equation is
max(mu)=max{1/sqrt(5), a/(a+2c)}.

I canít understand how to calculate max(mu). Anybody know? I have tried to analyze the eigenvalues of my iteration matrix, the one with maximal norm is about 0.005, however, the convergence rate in my actual numerical experiment is closed to 1/sqrt(5). How to calculate the convergence rate 1/sqrt(5) theoretically?

Thank you so much.
hadesmajesty is offline   Reply With Quote


Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On

Similar Threads
Thread Thread Starter Forum Replies Last Post
BlockMeshmergePatchPairs hjasak OpenFOAM Native Meshers: blockMesh 11 August 15, 2008 07:36
BlockMeshmergePatchPairs polyTopoChanger benru OpenFOAM Native Meshers: blockMesh 3 June 29, 2008 21:24
errors Fahad Main CFD Forum 0 March 23, 2004 14:20
Problems of Duns Codes! Martin J Main CFD Forum 8 August 14, 2003 23:19
error while compiling the USER Sub routine CFD user CFX 3 November 25, 2002 16:16

All times are GMT -4. The time now is 15:37.