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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)
  • References: <e8888u$92q$1@smc.vnet.net> <200607031038.GAA16538@smc.vnet.net> <acbec1a40607050108o4999ab27r6014aa907d8a66f4@mail.gmail.com> <Pine.LNX.4.58.0607051753490.3084@elmo.imtek.uni-freiburg.de>
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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/


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