On stability of difference schemes

TL;DR

This paper investigates the stability of nonlinear explicit difference schemes, introduces a second-order accurate modification of the Lax-Friedrichs scheme, and demonstrates its robustness and accuracy through tests on conservation laws.

Contribution

It establishes a stability criterion for nonlinear schemes and develops a new second-order nonstaggered central scheme with improved accuracy.

Findings

Modified scheme is stable and accurate on conservation laws

Scheme is robust for hyperbolic laws with stiff source terms

Second-order nonstaggered scheme outperforms traditional methods

Abstract

The stability of nonlinear explicit difference schemes with not, in general, open domains of the scheme operators are studied. For the case of path-connected, bounded, and Lipschitz domains, we establish the notion that a multi-level nonlinear explicit scheme is stable iff (if and only if) the corresponding scheme in variations is stable. A new modification of the central Lax-Friedrichs (LxF) scheme is developed to be of the second order accuracy. The modified scheme is based on nonstaggered grids. A monotone piecewise cubic interpolation is used in the central scheme to give an accurate approximation for the model in question. The stability of the modified scheme is investigated. Some versions of the modified scheme are tested on several conservation laws, and the scheme is found to be accurate and robust. As applied to hyperbolic conservation laws with, in general, stiff source terms, it is constructed a second order nonstaggered central scheme based on operator-splitting techniques.