Re: Re: finite differencing of a PDE system
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- Subject: [mg67763] Re: [mg67656] Re: finite differencing of a PDE system
- From: "Chris Chiasson" <chris at chiasson.name>
- Date: Thu, 6 Jul 2006 06:54:32 -0400 (EDT)
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Thank you, Gregor and Oliver Ruebenkoenig, again for your responses. They are most appreciated. I created an example notebook with a lot of explanatory text that somewhat illuminates the issues we've been talking about. http://chris.chiasson.name/temp/discretization_example.nb Here is a summary of my inferences: The use of a derivative differencing scheme that automatically switches from center difference to forward or backward differences as boundaries are approached creates systems of equations that are not (completely) linearly independent. The value of MatrixRank as applied to the output of CoefficientArrays for a given set of equations tells one how many equations are independent. The number of additional equations needed to achieve a rank equal to the number of unknowns is exactly equal to the number of required boundary values for problems with Dirichlet boundary conditions. At least two methods exist for obtaining the finite difference solutions. One of them involves augmenting the input to CoefficientArrays so that it produces a rectangular coefficient matrix. The least squares solution is obtained by the use of the pseudo-inverse of the rectangular coefficient matrix, because the extra dependant equations do not affect the solution much. The other method involves the use of imsDirichlet to eliminate the rank deficiency of the square coefficient matrix by inserting the boundary conditions into the final equation system. I am not sure exactly how it works, because it does more than simply change one of the rows of the augmented ( m|b ) matrix. The imsDirichlet method is a few orders of magnitude faster. The pseudo-inverse method is a few times closer to the exact solution. This should be enough research to attempt a solution to the larger problem on which I am working. Thank you for your input, -- http://chris.chiasson.name/
- References:
- Re: finite differencing of a PDE system
- From: Oliver Ruebenkoenig <ruebenko@uni-freiburg.de>
- Re: finite differencing of a PDE system