The Riemann problem for one dimensional generalized Chaplygin gas dynamics

Abstract

a b s t r a c tThe Riemann problem for one dimensional generalized Chaplygin gas dynamics is considered. Its two characteristic fields are genuinely nonlinear, but the nonclassical solutions appear. The formation of mechanism for δ-shock is analyzed, that is the oneshock curve and the two-shock curve do not intersect each other in the phase plane. The Riemann solutions are constructed, and the generalized Rankine-Hugoniot conditions and the δ-entropy condition are clarified. By the interaction of the delta-shock wave with the elementary waves, the generalized Riemann problem for this system is presented.Furthermore, by studying the limits of the solutions as perturbed parameter ε approaches zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data. Some numerical simulations are given to illustrate our analysis.Crown

The Riemann problem for the Chaplygin gas equations with a source term

“…If and with and instead of , then the system (1.1) becomes the so‐called one‐dimensional generalized Chaplygin gas equations. In fact, it was shown in that the delta shock wave also appears in the Riemann solutions to the generalized Chaplygin gas equations. It should be remarkable that a significant mathematical difference between the generalized Chaplygin gas equations and the Chaplygin gas equations for the reason that the characteristic fields are genuinely nonlinear for the generalized Chaplygin gas equations.…”

The delta-shock wave for the two variables of a class of Temple system

“…The data (1.5) is a perturbation of the Riemann initial data (1.4). By discussing the interactions of δ-shock and elementary waves, we demonstrate whether the Riemann solutions of (1.1) and (1.4) are the limits of the solutions of (1.1) and (1.5) as ε → 0; see also [18,28,36] and the references therein. In addition, we refer the readers to the monograph of Chang and Hsiao [3] or [30] for the work on interactions of elementary waves and the stability of the Riemann solutions.…”

The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

“…The Riemann solutions of equations (3) are constructed by the analysis method in phase plane. In 2013, Wang [5] studied the generalized Chaplygin equations with constant initial data and obtained the global Riemann solution involving nonclassical wave (delta shock wave). In 2016, Sun [6] considered system (3) with a source term and constructed the exact solutions which contain the delta shock wave.…”