poisson equation
 

Hi all,

I'm dealing with a problem since one month and finally I prefer to ask in order to have your point of view.

I have to solve a poisson equation, and I've tried many numerical methods : direct and iterative ones. Matrix are very large [800][800] is the minimum, and until now I still didn't find a solution. No one of the method I applied converged... Jacobi, Gauss-Seidel

So anyone have any suggestion about this.

Re: poisson equation
 

you might be making a mistake, which is difficult to tell from what you have posted, try reading how to discretise poisson equation from cfd-wiki (on cfd-online.com). I wrote small portion describing that. and yes, it converges well with gauss seidel etc.

Re: poisson equation
 

Make sure that the applied boundary conditions are correct.If you use a tridiagonal solver with implicit scheme check for diagonal dominance.What kind of source terms and boundary conditions do you apply ?

A simple mathematical statement might help the cause :)

Re: poisson equation
 

First of all, make sure that your solver is correctly implemented and that you're using it correctly. It can be easily done by using a short and "controled" system (one that you have the solution) of equations (let's say 5x5 or 3x3). After that, if your method keep diverging you can go to check your Poisson discrete solution. As zxaar said, the wiki page can help you in this phase.

Cheers

Renato.

Re: poisson equation
 

To make sure, your code is working fine, Benchmark your code with simple problems like. cavity flow or backward step facing flow etc...

Re: poisson equation
 

Well, first of all, thanx for considering my post.

I'll try to answer all the question you've asked in order to make the situation clear.

I'm performing incompressible planar bidimensional and laminar Navier Stokes solver. I've derived consistant pressure equation (laplacian) and boundaries conditions (newmann).

I verified linear system Matrix that I reduced to [n][5] and vector construction.

The only problem is the method. I've read somewhere that sometimes a method don't work with in some specific cases.

I'm really discouraged, I've verified so many times all the algorithms.

Concerning Gauss Seidel and Jacobi, I've developped them before this project and I'm sure they are working. Actually I'm trying the SOR by lines and it seems to give logic results. (I didn't have any material providing pressure results to compare with) But its take long time to converge (8 min for a [2500][5] for a 10-4 eps)!!!

I have another question : do you think conjugate gradient method can be used in this case? It is really fast?

Re: poisson equation
 

Hi!

Gauss-Seidel and Jacobi are very old algorithms for iterative solution of linear equation systems. They have a rather bad convergence rate for big sparse matrices (and your equation system is big). Better algorithms are for example the SIP-algorithm or cgstab-family algorithms. If you want to use gauss-seidel or jacobi then only in combination with multigrid methods which are very powerful and have a fast convergence behaviour independent of size of matrices- they have optimal convergence behaviour. Google... try the book of ferziger and peric for a compact introduction of linear equation system solvers. you can also download many codes there... good luck!