Frozen coefficient method (von Neumann stability analysis)

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?

• In case the question gets little attention here, it might be useful to try scicomp.stackexchange.com – Igor Khavkine Nov 10 '14 at 23:14
• 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. – Michael Renardy Nov 11 '14 at 0:35
• @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. – cool Nov 11 '14 at 17:05
• Could you please give a page number for the particular statement in Gustafsson that you are referring to? – David Ketcheson Nov 16 '14 at 11:39
• 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 – cool Nov 16 '14 at 21:03