Poisson Equation Neumann boundaries singularity

  • #1
I am trying to solve the poisson equation with neumann BC's in a 2D cartesian geometry as part of a Navier-Stokes solver routine and was hoping for some help.

I am using a fast fourier transform in the x direction and a finite difference scheme in the y. This means the poisson equation becomes

-kx^2p_{i,j}+(((p_{i,j-1})-(2*p_{i,j})+(p_{i,j+1}))/(dy^2))=RHS

with dp/dy=0 at the boundaries being enforced using ghost boundary points

this inverts easily for all wavenumbers except when kx=0 when the matrix is singular.

I was wondering if anyone had any experience dealing with this problem and any standard methods of solving such a problem?

Can you simply set p_{hat}=0 at these points and then convert back to real space?

Thanks
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
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Is p_{i,j} the fourier transform? You should take care that the method does not produce some unwanted complex phase which might ruin the solution.
The kx=0 entry is the mean value for that "row" so you can not just set it equal to zero. I would imagine it depends a bit on your RHS how to best proceed.

PS: If you want to keep it fully spectral you could solve in the y-direction using a cosine transform.
 

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