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

摘要

This article deals with relaxation approximations of nonlinear systems ofhyperbolic balance laws. We introduce a class of relaxation schemes andestablish their stability and convergence to the solution of hyperbolic balancelaws before the formation of shocks, provided that we are within the frameworkof the compensated compactness method. Our analysis treats systems ofhyperbolic balance laws with source terms satisfying a special mechanism whichinduces 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 moregeneral source terms. The rate of convergence of the relaxation system to asolution of the balance laws in the smooth regime is established. Our workfollows 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, SIAMJ. on Num. Anal. (2005), 42-4] for systems of hyperbolic conservation lawswithout source terms.