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June 18, 2004, 11:54 
Poisson solver

#1 
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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 nonstaggered 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 pointbypoint 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!! 

June 18, 2004, 14:56 
Re: Poisson solver

#2 
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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 biconjugate gradient method.


June 24, 2004, 04:20 
Re: Poisson solver

#3 
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Ok, great, thanks! Is it necesary to include a preconditioner?


June 24, 2004, 14:31 
Re: Poisson solver

#4 
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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.


June 28, 2004, 06:06 
Re: Poisson solver

#5 
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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... 

June 28, 2004, 11:34 
Re: Poisson solver

#6 
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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 5pt or 7pt scheme, for my case I find that a tolerance of 1E8 and 1E10 do not differ much in the solutions. 

July 22, 2004, 21:16 
Re: Poisson solver

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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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