The Riemann problem and wave interactions for a relativistic $p$-system with variable heat ratios

We comprehensively investigate the Riemann problem for a relativistic $p$-system with variable heat ratios. First, we consider the relativistic system when the heat ratio $\gamma$ changes. We supplement with the trivial equation $\frac{\mathrm{\partial }}{\mathrm{\partial }t}\gamma =0$ to close the system. Second, we solve the Riemann problem with piecewise constant of heat ratios, and obtain the elementary waves, including rarefaction waves, shock waves, contact discontinuities, and particularly stationary waves. An admissible stationary wave selection method based on monotone criterion is studied. Finally, we discuss the interaction of elementary waves, particularly, rarefaction wave and shock wave interactions with stationary wave, respectively. The large-time behavior of each case is also presented.