The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws

Guodong Wang,Lin Zhao,Jia-Bao Liu

doi:10.1186/s13661-017-0837-2

Abstract

The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws is considered. By the properties of wave curves, we observe that this system does not contain the composite wave compared to the barotropic models of gas dynamics with different pressure laws. Under some initial value data, the Riemann solution is constructed. Using the interaction of the elementary waves, we consider the generalized Riemann problem and discover that the Riemann solution is stable for such perturbation of the initial data.

Highlights

1 Introduction One-dimensional nonlinear wave system with the variable gamma laws which only depends on the spatial coordination is described as follows:
System ( . ) can be deduced from the barotropic models of gas dynamics with different pressure laws in [ ], Wang et al Boundary Value Problems (2017) 2017:107 and we refer the reader to this paper for details
Which is similar to the model of a mixture of gases governed by different gamma laws in

Summary

Introduction

One-dimensional nonlinear wave system with the variable gamma laws which only depends on the spatial coordination is described as follows:. ) can be deduced from the barotropic models of gas dynamics with different pressure laws in [ ], Wang et al Boundary Value Problems (2017) 2017:107 and we refer the reader to this paper for details. We observe that the corresponding Riemann solution is similar to that of the model of onedimensional adiabatic flow in Lagrangian coordinates, we see the related results [ – ] for details. In Section , we consider the initial value problem with three constant states. By the interaction between the stationary contact wave and the shock wave or rarefaction wave, the global solutions are constructed. We obtain that the solution of the perturbed initial value problem converges to the corresponding Riemann solution as ε approaches zero, which shows the stability of the Riemann solution for the small perturbation

The solution to the Riemann problem

Concluding remarks