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?
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.