On monotonicity, stability, and construction of central schemes for hyperbolic conservation laws with source terms (Revised Version)

TL;DR

This paper develops new theoretical criteria for the stability and monotonicity of central schemes for hyperbolic conservation laws with source terms, introduces a second-order accurate modification of the Lax-Friedrichs scheme, and demonstrates its robustness through numerical tests.

Contribution

It presents a novel approach linking non-linear scheme stability to linear scheme variations, generalizes Friedrichs' theorem, and introduces a second-order central scheme with monotone cubic interpolation.

Findings

The modified scheme is accurate and robust.

Stability criteria are established for schemes with source terms.

Numerical tests confirm improved accuracy and stability.

Abstract

The monotonicity and 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 and monotonicity of a non-linear scheme in terms of its corresponding 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. The main methodological innovation is the theorems establishing the notion that a non-linear scheme is stable (and monotone) if the corresponding scheme in variations is stable (and, respectively, monotone). Criteria are developed for monotonicity and stability of difference schemes associated with the numerical analysis of systems of partial differential equations. The theorem of Friedrichs (1954) is generalized to be applicable to variational schemes with non-symmetric matrices. 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 and monotonicity 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 scheme based on operator-splitting techniques.