The relaxation schemes for systems of conservation laws in arbitrary space dimensions

TLDR

The paper introduces a class of relaxation schemes for solving systems of conservation laws in multiple spatial dimensions. These schemes employ a local relaxation approximation that constructs a linear hyperbolic system with a stiff lower‑order term, allowing stable under‑resolved discretizations without the need for Riemann solvers or nonlinear algebraic solvers. Numerical experiments in one and two dimensions demonstrate that the second‑order schemes are total‑variation diminishing in the zero‑relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.

Abstract

Abstract We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1‐D and 2‐D problems are presented. The second‐order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.