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Poisson equation: non-symmetric matrix with non-uniform spacing?

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Old   July 25, 2012, 03:22
Default Poisson equation: non-symmetric matrix with non-uniform spacing?
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Daniel Weber
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Hello,

I'm trying solve a poisson equation with finite differences. According to "Computational Methods for Fluid Dynamics" the finite difference approximation of the laplace operator in the 1D poisson equation is

p_i+1 (x-i - x_i-1) + p_i-1(x_i+1 - x_i) - p_i (x_i+1 - x_i) / [ 1/2 * (x_i+1 - x_i-1) * (x_i+1 - x_i ) * ( x_i - x_i-1) ]

If the grid spacing is uniform the matrix is symmetric, but not for non-uniform grid spacing. With non-uniformity the requirements is that the distance between each pair of odd and even unknowns (x_i - x_i+2, x_i+1 - x_i+3) must be equal. I could artificially move the unknows a bit, but generally that limits the transformation function of the grid. I would like to use a CG method, so is there anything I can do modify the scheme. (or maybe use finite volumes?)

How is the pressure correction done in Finite Volume methods? Are the matrices as well non-symmetric? Any hint or a pointer to literature would be very helpful!

Thank you!
Daniel
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Old   July 25, 2012, 03:54
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Filippo Maria Denaro
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Quote:
Originally Posted by megaweber View Post
Hello,

I'm trying solve a poisson equation with finite differences. According to "Computational Methods for Fluid Dynamics" the finite difference approximation of the laplace operator in the 1D poisson equation is

p_i+1 (x-i - x_i-1) + p_i-1(x_i+1 - x_i) - p_i (x_i+1 - x_i) / [ 1/2 * (x_i+1 - x_i-1) * (x_i+1 - x_i ) * ( x_i - x_i-1) ]

If the grid spacing is uniform the matrix is symmetric, but not for non-uniform grid spacing. With non-uniformity the requirements is that the distance between each pair of odd and even unknowns (x_i - x_i+2, x_i+1 - x_i+3) must be equal. I could artificially move the unknows a bit, but generally that limits the transformation function of the grid. I would like to use a CG method, so is there anything I can do modify the scheme. (or maybe use finite volumes?)

How is the pressure correction done in Finite Volume methods? Are the matrices as well non-symmetric? Any hint or a pointer to literature would be very helpful!

Thank you!
Daniel

Yes, on non-uniform grids the matrix is symmetric in its shape but not in the entry values
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Old   July 25, 2012, 08:21
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Yes, on non-uniform grids the matrix is symmetric in its shape but not in the entry values
Hi FMDenaro, can you be more specific when you say symmetric in its shape but not in the entry values? Im not an expert in this matter and now Im having a trouble using BICGstab with ILU preconditioning (MATLAB BICGstab and ILU functions). The ILU drops ill conditioned preconditioners according to BICGstab warning, also the ILU function shows a warning about getting pivots with zero value. This is for an inlet outlet problem using CVFEM on triangular grid (the code works fine on lid driven cavity flow).

Best regards...
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Old   July 26, 2012, 03:03
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Poisson operator should give you symmetric matrix ie
Aij = Aji

It is true in case of non uniform meshes too.
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Old   July 26, 2012, 03:10
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Filippo Maria Denaro
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Poisson operator should give you symmetric matrix ie
Aij = Aji

It is true in case of non uniform meshes too.

I dont think so ... consider for example a 1D example in [0, L] with a stretched grid near L.
Symmetry remains in the pattern, not in values
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Old   July 26, 2012, 05:52
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I dont think so ... consider for example a 1D example in [0, L] with a stretched grid near L.
Symmetry remains in the pattern, not in values
checked and Agreed. it seems in FD it does comes out to be unsymm system.
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Old   July 26, 2012, 11:16
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Daniel Weber
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Hi,

thanks for the answers. Yes, I meant non-symmetric in the values.

But I'm still curios if there are as well non-symmetric matrices(in the values) when I use a finite volume discretization for the pressure correction. Does anybody have i hint?

best regards,
Daniel
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