On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

This paper is concerned with the four-wave Riemann problem for a two-dimensional hyperbolic system of nonlinear conservation laws derived from a quasi-linear wave equation. The self-similar form of this system is of mixed type. The four-wave Riemann problem in the self-similar plane consists of interactions of four planar elementary waves (exterior waves), which contain rarefaction waves, shocks and contact discontinuities. The Riemann problem is classified into sixteen genuinely different nontrivial cases. The structures of solutions for four rarefaction waves, four shocks and two nonadjacent rarefaction waves plus two nonadjacent shocks are constructed completely. For the rest cases, the solutions are roughly analyzed. For each case, the corresponding numerical solutions are illustrated via contour plots. Comparing with the compressible Euler equations and related models, one of the highlights for this paper is that the interactions of two rarefaction waves, two shocks, as well as a rarefaction wave and a shock in hyperbolic domains are clarified.