Riemann problem for the isentropic Chaplygin gas Cargo-LeRoux model

Abstract

The so-called Cargo-LeRoux model is a hyperbolic system of conservation laws derived from the Euler equations with a constant gravity. This paper is concerned with such a model for an isentropic Chaplygin gas. The Riemann problem is solved with four kinds of structures. The first is composed of the contact discontinuities, the second contains an overcompressible δ-wave (δ-shock wave), and the third and fourth involve a contact discontinuity and a kind of non-overcompressible δ-wave. One of the highlights of this paper lies in the introduction of the non-overcompressible δ-waves.

Embedded delta shocks

“…Here we have constructed an elegant solution comprising two -shocks. In [19], a solution with three -shocks appears for a 3 × 3 system of conservation laws. Actually, following the ideas in Sec.…”

“…Of course, these computations hold in the sense of distributions, see [19,21,22]. However, the Riemann solutions in the sections that follow comprise rarefactions that are difficult to handle in these distributions.…”

“…However, the Riemann solutions in the sections that follow comprise rarefactions that are difficult to handle in these distributions. Even if it is possible to compute the generalized Rankine-Hugoniot conditions given in [6], see also [19], for simplicity we prefer direct computations as in (9).…”

“…Remarkably, the solutions we present here possess a -shock rather than these intermediate constant states. Other directions are given in [19], where Riemann solutions are reported that possess no intermediate constant states but -contact discontinuities and, in [5], where interaction of classical waves and -shocks is given at a positive time.…”

Pressureless Magnetohydrodynamics System: Riemann Problem and Vanishing Magnetic Field Limit

“…For instance, Li and Chen and Liu discussed this topic by considering the vanishing pressure limits of solutions to the isentropic [20,21] and nonisentropic Euler equations [22]; Mitrović and Nedeljkov [23] discussed this topic by perturbing the generalized pressureless gas dynamics model; Shen and Sun [24] discussed this topic by studying the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model; Cheng and Yang [25] discussed this topic by investigating the partly vanishing pressure limits of solutions to a nonsymmetric Keyfitz-Kranzer system of conservation laws with generalized and modified Chaplygin gas; Yin and Sheng [26,27] discussed this topic by considering the vanishing pressure limits of solutions to the relativistic Euler equations; Yang and Liu [28,29] discussed this topic by introducing some flux approximations in the isentropic and nonisentropic classical Euler equations; and Sahoo and Sen [30] discussed this topic by considering the limiting behavior of two strictly hyperbolic systems of conservation laws. Compared with the delta shocks, we also refer the readers to [31][32][33] for δ n -shocks and [34] for noncompressible δ-waves.…”