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Poisson equation: nonsymmetric matrix with nonuniform spacing? 

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July 25, 2012, 03:22 
Poisson equation: nonsymmetric matrix with nonuniform spacing?

#1 
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Daniel Weber
Join Date: Nov 2009
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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 (xi  x_i1) + p_i1(x_i+1  x_i)  p_i (x_i+1  x_i) / [ 1/2 * (x_i+1  x_i1) * (x_i+1  x_i ) * ( x_i  x_i1) ] If the grid spacing is uniform the matrix is symmetric, but not for nonuniform grid spacing. With nonuniformity 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 nonsymmetric? 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:
Yes, on nonuniform 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:
Best regards... 

July 26, 2012, 03:03 

#4 
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Arjun
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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 

#5 
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Filippo Maria Denaro
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July 26, 2012, 05:52 

#6 
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Arjun
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July 26, 2012, 11:16 

#7 
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Daniel Weber
Join Date: Nov 2009
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Hi,
thanks for the answers. Yes, I meant nonsymmetric in the values. But I'm still curios if there are as well nonsymmetric 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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