Poisson equation: non-symmetric matrix with non-uniform spacing?
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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Yes, on non-uniform grids the matrix is symmetric in its shape but not in the entry values |
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Best regards... |
Poisson operator should give you symmetric matrix ie
Aij = Aji It is true in case of non uniform meshes too. |
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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 |
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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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