On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. I. Nonstiff strongly dynamic problems

Abstract

An implicit finite difference method of fourth order accuracy (in space and time) is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of this method is a straightforward generalization of the Crank-Nicholson scheme: it is a two-level scheme which is unconditionally stable and nondissipative. The scheme is compact, i.e., it uses only 3 mesh points at level t and 3 mesh points at level t + Δ t t + \Delta t . In this paper, the first in a series, we present a dissipative version of the basic method which is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are presented to illustrate properties of the proposed scheme.