Stability and Convergence of Relaxation Schemes to Hyperbolic Balance Laws via a Wave Operator

TL;DR

This paper introduces relaxation schemes for hyperbolic balance laws, proving their stability and convergence before shock formation, and analyzes the convergence rate in smooth regimes within the compensated compactness framework.

Contribution

It presents a new class of relaxation schemes for hyperbolic balance laws with source terms, establishing their stability, convergence, and convergence rates in the smooth regime.

Findings

Relaxation schemes are stable and converge to solutions before shock formation.

Convergence rates are established for smooth solutions.

The analysis extends to systems with weakly dissipative source terms.

Abstract

This article deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [C.M. Dafermos J. Hyp. Diff. Equations, 3, 505-527, 2006], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented in [S. Jin, X. Xin, Comm. Pure. Appl. Math. (1995), 48] and [Ch. Arvanitis, Ch. Makridakis, and A.E. Tzavaras, SIAM J. on Num. Anal. (2005), 42-4] for systems of hyperbolic conservation laws without source terms.