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Old   June 18, 2004, 12:54
Default Poisson solver
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Steve
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Hi, I am writing a code for solving viscous incompressible flows using Navier Stokes equations in (u,v,p). Using normal cartesian non-staggered grid.

I am using the fractional step method, so I will be solving for intermediate velocities and then a pressure Poisson equation to get the new velocities.

Presently I am using the point by point SOR method for my Poisson solver. I would like to ask what Poisson solution method would be better, if I am still required to solve on point-by-point basis. I am thinking of using conjugate gradient, and for my code, multigrid method is not advisable.

Another thing is, for conjugate gradient, do I need to include my boundary equations in Ax=b ??

Thanks a lot!!
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Old   June 18, 2004, 15:56
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You will have to include boundary equations in Ax=b. This will make your matrix A asymmetric. For this you should use the bi-conjugate gradient method.
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Old   June 24, 2004, 05:20
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Steven
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Ok, great, thanks! Is it necesary to include a preconditioner?
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Old   June 24, 2004, 15:31
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Preconditioning is used to accelerate convergence. If your convergence is slow it would be worthwhile using it since you will be solving Poisson equation in every time step.
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Old   June 28, 2004, 07:06
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Steve
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Any recomendations for a preconditioner? I tried without one and one with the Jacobi precond under NEUMANN boundary conditions for a Poisson problem of f=cos(0.5pi x)cos(0.5pi y). But I couldn't derive machine zero convergence for both methods. Any particular reason why?

I am using a meshfree method, so it would be advisable for me to implement a precond that doesn't require the transpose of the A matrix (in Ax=b) or preferrably something less fanciful...
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Old   June 28, 2004, 12:34
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Since i haven't written any preconditioners myself, I'm sorry I cannot help you in this part. But there should be books on it for help.

Regarding machine accuracy, it takes too many iterations to reach, also I cannot reach it most of the time. For second order 5-pt or 7-pt scheme, for my case I find that a tolerance of 1E-8 and 1E-10 do not differ much in the solutions.
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Old   July 22, 2004, 22:16
Default Re: Poisson solver
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cregeo
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May I add one question to this thread?

For the poisson equation that will turn out

to be matrix equation:

div(grad(p) = f

how can I insert the Neumann boundary conditions into the matrix:

<grad(p), n> = 0

where <,> is inner product and n is a unit normal on the boundary.

In a staggered grid, the boundary condition turns out to be as follows usually:

p_i = (- n_x * p_(j))/(n_x + n_y) + (-n_y * p(k))/(n_x + n_y)

where p_i, p_j, and p_k are the pressure at cell centers and neighbor to each other..

My question is whether I should replace this into the poisson equation modifying the poisson equation without changing the size of the matrix or use the boundary condition as a Lagrange Multiplier?

Please help me..

Cregeo
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