Analysis of the Stability of the Riemann Problem for a Simplified Model in Magnetogasdynamics

Yujin Liu,Wenhua Sun

doi:10.1155/2016/7526413

Abstract

The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin(t>0)in the(x,t)plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data.

Highlights

Magnetogasdynamics plays an important role when we study engineering physics and many other aspects and is important for the hyperbolic theory
We find that the contact discontinuity appears after perturbation while the corresponding Riemann solution has no contact discontinuity for some case
The perturbation can affect the corresponding Riemann solution which shows that the corresponding Riemann solution is unstable

Summary

Yujin Liu and Wenhua Sun

The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin (t > 0) in the (x, t) plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. For some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. The Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. For some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data

Introduction

Advances in Mathematical Physics

The contact discontinuity can be expressed as

It follows that the Riemann solution is

And we get the