Delta-shocks and vacuums in a geometrical optics system by the flux approximations

In this paper, we construct three classes of flux-approximation models to analyze the formation phenomena of delta shock waves and vacuum states in a geometrical optics system. Based on the obtained Riemann solutions of these models, we prove that, when the single, double, and triple parameters vanish, respectively, any one-parameterized-delta-shock, one-shock-wave and one-contact-discontinuity, two-shock-wave Riemann solutions of the flux-approximation models converge to a (right contact) delta-shock solution of the geometrical optics system; any one-rarefaction-wave and one-contact-discontinuity, one-rarefaction-wave, two-rarefaction-wave Riemann solutions of the flux-approximation models tend to a vacuum solution of the geometrical optics system. In addition, we also present some representative numerical simulations which completely illustrate the theoretical analysis.