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

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.