Dear Paresh, 

From what equations does your matrix come from? Have you looked at the eigenvalues of your matrix?  I ask you these questions because Gauss-Seidel (and Jacboi) only work for symmetric positive definite (SPD) matrices.  This is also true for conjugate gradient (CG). In fact, most theories on convergence of iterative methods are based on SPD matrices.  Boundary conditions may break symmetry for you so this leaves you with moving to GMRES  or BiCGStab.  GMRES  is nothing more than the generalization of CG (where instead of using Lanczos iteration, the Arnoldi iteration is used which is outlined in texts such as Trefethen and Bau).  BiCGStab will also work but it requires two applications of the system matrix (instead of one for GMRES) so we have never seen BiCGStab beat GMRES in efficiency.  I wouldn't worry about preconditioning just now - once you get this to work, then you can worry about efficiency and I would be happy to propose preconditioners to you at that point. 

The first thing you need to do is diagnose why Gauss-Seidel is not working. Can you run a small version of your problem so that you can actually store a small version of your system matrix? If so, then analyze it with, say, Matlab, and get the eigenvalues of your system.  If they are all real and positive, and symmetric then Gauss-Seidel, Jacobi,and CG should work. If they are non-symmetric but real and positive then GMRES will work like a charm. If they are not real and positive, then you will need preconditioning for sure and I can show you how to do this using simple techniques such as polynomial preconditioning.  hope this helps.

try BiCGSTAB its a combination of BiCG and GMRES.

In whole, the implicit numerical methods are more stable than the explicit. The reason for that is generally due to the Courant number. It is better to check your way implementing boundary condition. Moreover, you can utilize the methods, like SOR, other than Gauss Siedel.

You've to see if the jacobian matrix is symmetric or not. ConjGrad based schemes are fast and more appropriate for symmetric matrices, but GMRES is better off for asymmetric ones. If the matric is not too big, using non-stationary iterative methods these that I mentioned may not be required, stationary ones (SGS, LU-SGS, PJ) etc. are good enough.

@Aldo Dall'Osso Sir,

I am already using under-relaxation parameter of 0.2-0.9 to enable code to run..

But every time, it does not work.

Paresh,

Don't quite understand what problem you have. Do you use pressure-velocity approach? Or hyperbolic system? The "instability" problem occurrs at what stage?

Dear Paresh, 

From what equations does your matrix come from? Have you looked at the eigenvalues of your matrix?  I ask you these questions because Gauss-Seidel (and Jacboi) only work for symmetric positive definite (SPD) matrices.  This is also true for conjugate gradient (CG). In fact, most theories on convergence of iterative methods are based on SPD matrices.  Boundary conditions may break symmetry for you so this leaves you with moving to GMRES  or BiCGStab.  GMRES  is nothing more than the generalization of CG (where instead of using Lanczos iteration, the Arnoldi iteration is used which is outlined in texts such as Trefethen and Bau).  BiCGStab will also work but it requires two applications of the system matrix (instead of one for GMRES) so we have never seen BiCGStab beat GMRES in efficiency.  I wouldn't worry about preconditioning just now - once you get this to work, then you can worry about efficiency and I would be happy to propose preconditioners to you at that point. 

The first thing you need to do is diagnose why Gauss-Seidel is not working. Can you run a small version of your problem so that you can actually store a small version of your system matrix? If so, then analyze it with, say, Matlab, and get the eigenvalues of your system.  If they are all real and positive, and symmetric then Gauss-Seidel, Jacobi,and CG should work. If they are non-symmetric but real and positive then GMRES will work like a charm. If they are not real and positive, then you will need preconditioning for sure and I can show you how to do this using simple techniques such as polynomial preconditioning.  hope this helps.

Dear Paresh,

Please identify the type of flow you are trying to solve and the type of stencil you are using. Depending on the type of system, you may not be able to use certain discretization schemes. Are you solving hyperbolic equations? Are you using upwinding? Can you provide a little more information perhaps you can scan a page with the expression of the equations you are trying to solve?

Best Regards,

Timur

as mentioned, there is a bunch of robust num. algo. for solving linear systems. CG, BiCGSTAB, GMRS...etc. The choice of the num. algo depends on the system of equations you deal with. Beside the mentioned algorithms, keep in minde that the preconditioner are very important in order to guarantee a fast convergence.

It will be helpful if you can identify the type of equations you deal with

cheers

Gauss Seidel Iterative method to solve CFD equations is one of the iterative numerical methods. The following link discusses the method and the convergence of the iteration methods, see page 46 and up at the link:

https://www.math.ust.hk/~mamu/courses/231/Slides/CH07_3B.pdf

This is a good question.

In addition to the incisive answers already given, consider

J. Qian, Y.-T. Zhang, H.-K. Zhao, A fast sweeping method for static convex Hamiltonian-Jacobi equations, report, Wichita State University and University of California Irvine:

ftp://ftp.math.ucla.edu/pub/camreport/cam06-37.pdf

See Section 1, page 2, where a fast sweeping strategy is used to solve a nonlinear boundary value problem and which gives an efficient, unconditionally stable iteratiive method.

There are two different things you concern in your question. If you mean the numerical stability w.r.t. the rounding errors (i.e. the cancellation phenomena), than it is quite hard to eliminate them by the choice of the solver. These phenomena arise usually if the mean pressure is very high in comparison with its variations (and you cannot shift it due to the model or boundary conditions) so that the finite differences approximating the gradient of the pressure (in any discretization) suffer of the loss of precision. Then the norm of the defect cannot be reduced below some value, and this is not due to the solver but due to the computation of the defect and the correction. But then I would expect the oscillation of the norm of the defect only in the equation where the gradient of the pressure occur.

If you mean really the convergence properties of the solver itself, you should really specify more details about your CFD problem and its discretization. Without it, you will get only very general answers about the symmetry, the eigenvalues, the CG-like methods etc. One should know at least, at which stage your GS method is used. There is a difference, whether this are full equations or some part of them (fixed pressure, ...); a compressible or an incompressible case, if incompressible, then Boussinesq, or what; convection-dominant, or very viscous and diffusive (for the temperature); stationary or time-dependent, and in the latter case, whether you solve the non-linear system, or the linearization is included in the time-stepping scheme; a collocated spatial discretization scheme (for ex., with a stabilization), or not. Krylov-type methods, like BiCGStab, would be worth any way, but surely only if you choose a good preconditioner. If you mean, that your preconditioner is a geometric multigrid method (this would be more or less a modern choice) with the GS smoother, and you observe the bad convergence, than the convective terms could be the reason. Then I would try ILU, possibly with a parameter between 0 and 1. But the linearization may have an influence, too. Robustness of linear solvers is a big problem in CFD, so it is quite complicated to find a generally "stable" (rapidly converging) solver. The diagonal dominance (often together with the sign inequality, like for the M-matrices) can provide general applicability and the fact of convergence of a method, but usually says nothing about the convergence rate.

Note that the two problems are of different natures, and both can be present, but should be solved by different means.

You can found  what you want in one of the best Book on this subjet :

Iterative Methods for sparse Linear Systems

by Yousef Saad

And it is free to download ...

You can also find differents library  of iterative solvers too

http://www-users.cs.umn.edu/~saad/IterMethBook_2ndEd.pdf

http://www-users.cs.umn.edu/~saad/software/

Gauss Seidel and Jacobi are stable only if your system is diagonally dominant. Try using better methods like BiCGSTAB, GMRES etc...

http://www.netlib.org/templates/templates.pdf