The Perturbed Riemann Problem for a Geometrical Optics System

Abstract

The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved. By studying the interactions among of the delta-shock, vacuum, and contact discontinuity, fourteen kinds of structures of Riemann solutions are obtained. The compound wave solutions consisting of delta-shocks, vacuums, and contact discontinuities are found. The single and double closed vacuum cavitations develop in solutions. Furthermore, it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data. Finally, the numerical results completely coinciding with theoretical analysis are presented.