On stability of difference schemes. Central schemes for hyperbolic conservation laws with source terms

TL;DR

This paper investigates the stability of difference schemes for hyperbolic conservation laws with source terms, introducing a second-order accurate central scheme with stability analysis and testing its robustness and accuracy.

Contribution

It develops a new second-order nonstaggered central scheme with stability analysis based on variations, and applies it to hyperbolic laws with stiff source terms.

Findings

The modified scheme is stable and accurate.

The scheme performs well on several conservation laws.

It is robust and suitable for stiff source terms.

Abstract

The stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability of a non-linear scheme in terms of its cor- responding scheme in variations. Such an approach leads to application of the stability theory for linear equation systems to establish stability of the corresponding non-linear scheme. It is established the notion that a non-linear scheme is stable 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. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The stability of the modified scheme are 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.