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

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 July 25, 2012, 03:22 Poisson equation: non-symmetric matrix with non-uniform spacing? #1 New Member   Daniel Weber Join Date: Nov 2009 Posts: 4 Rep Power: 10 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

July 25, 2012, 03:54
#2
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Filippo Maria Denaro
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Quote:
 Originally Posted by megaweber 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

July 25, 2012, 08:21
#3
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G. S.
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Quote:
 Originally Posted by FMDenaro 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? I´m not an expert in this matter and now I´m 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...

 July 26, 2012, 03:03 #4 Senior Member   Arjun Join Date: Mar 2009 Location: Nurenberg, Germany Posts: 733 Rep Power: 19 Poisson operator should give you symmetric matrix ie Aij = Aji It is true in case of non uniform meshes too.

July 26, 2012, 03:10
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Filippo Maria Denaro
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Quote:
 Originally Posted by arjun 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

July 26, 2012, 05:52
#6
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Arjun
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Quote:
 Originally Posted by FMDenaro 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.

 July 26, 2012, 11:16 #7 New Member   Daniel Weber Join Date: Nov 2009 Posts: 4 Rep Power: 10 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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