Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by Gustafsson,Kreiss,Oliger that there are some conditions, which give us possibility to use frozen coefficients+Neumann analysis at the level of full rigor. As a book didn't give any reference, could you help me to find these conditions?
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$\begingroup$ In case the question gets little attention here, it might be useful to try scicomp.stackexchange.com $\endgroup$– Igor KhavkineCommented Nov 10, 2014 at 23:14
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1$\begingroup$ This question, if I understand it correctly, seems to open a rather big can of worms. Before you launch into finite differences, you might want to consider what happens at the PDE level. For the "standard" types of PDEs, well-posedness of variable coefficient problems can be deduced from that of constant coefficient problems, but this notoriously fails for more "exotic" types of PDEs, e.g. hydrostatic approximation of fluid mechanics. $\endgroup$– Michael RenardyCommented Nov 11, 2014 at 0:35
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$\begingroup$ @MichaelRenardy, actually, this question can be asked both for well-posed and ill-posed problems. Gustafsson and others consider mainly hyperbolic equations and corresponding IBVPs. My main interest is connected with parabolic equations, for which there is nothing exact in frozen coefficient method. $\endgroup$– coolCommented Nov 11, 2014 at 17:05
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$\begingroup$ Could you please give a page number for the particular statement in Gustafsson that you are referring to? $\endgroup$– David KetchesonCommented Nov 16, 2014 at 11:39
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$\begingroup$ This is not a statement, this is just a piece of text at the page 183: "With certain extra conditions, it is possible to prove stability for the variable coefficient problem if all the "frozen" problems are stable. This has been done for approximations of hyperbolic and parabolic equations, and the results for hyperbolic equations will be given in Sections 6.5 and 6.6." Bertil Gustafsson, Heinz-Otto Kreiss, Joseph Oliger Time Dependent Problems and Difference Methods Pure and Applied Mathematics A Wiley-Interscience Series of Texts, Monographs and Tracts 1996 $\endgroup$– coolCommented Nov 16, 2014 at 21:03
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I believe the intended reference regarding parabolic PDEs is:
Fritz, John. On integration of parabolic equations by difference methods: I. Linear and quasi-linear equations for the infinite interval. Communications on Pure and Applied Mathematics, 5(2):155--211 (1952).
The paper is 57 pages long and I cannot find a later reference that neatly summarizes it. The nonlinear case is handled in the last section (Section 8).
I found this paper while looking through the references in
Strang, Gilbert. Accurate partial difference methods II. Numerische Mathematik 6, 37-46 (1964).
which contains results for the hyperbolic case.
