Volume 27, pp. 78-93, 2007.

On difference schemes for quasilinear evolution problems

Rita Meyer-Spasche

Abstract

We review several methods leading to variable-coefficient schemes and/or to exact difference schemes for ordinary differential equations (error elimination; functional fitting; Principle of Coherence). Necessary and suffient conditions are given for $t$-independence of fitted RK coefficients. Conditions for $\tau$-independence are investigated, $\tau$ the time-step. The theory is illustrated by examples. In particular, examples are given for non-uniqueness of exact schemes and for efficient difference schemes based on exact schemes and well suited for highly oscillatory ordinary differential systems or for parabolic equations with blow-up solutions.

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Key words

difference schemes, time stepping, nonstandard schemes, exact schemes, exponential fitting, functional fitting, Runge-Kutta, collocation methods, review

AMS subject classifications

65L05, 65M06, 65P99