Flux-approximation limits of solutions to the Brio system with two independent parameters

Abstract

By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a simplified system of conservation laws derived from the perturbed Brio system by neglecting some quadratic term. As one of the purebred parameters of the Brio system goes to zero, the solution of which consisting of two shock waves tends to a delta-shock solution to this simplified system. By contrast, the solution containing two rarefaction waves converges to a contact discontinuity and a rarefaction wave of the simplified system. What is more, the formation mechanisms of delta shock waves under flux approximation with both two parameters and only one parameter are clarified.