Exact solution of the flux perturbed Riemann problem for Cargo-LeRoux model in a van der Waals gas

Abstract

In this paper, we consider the pressureless Cargo-LeRoux model of conservation laws represented by a system of quasi-linear partial differential equations derived from the one-dimensional Euler equations with constant gravity. Considering flux perturbation of a van der Waals gas equation of state, we derive the exact solution of this Riemann problem based on the elementary wave analysis including shock wave, rarefaction wave and contact discontinuity wave. The Newton-Raphson method of two variables is applied to find the densities across the contact discontinuity wave of two nonlinear algebraic equations in all possible combinations of waves. Finally, this study exclusively reveals the influence of the growth of van der Waals excluded volume on the physical quantities: density, potential, velocity and total pressure through a series of test cases.