Numerical study of loss of hyperbolicity using a cold plasma model

TL;DR

This paper investigates how a cold plasma model can lose hyperbolicity due to electron-ion collision effects, proposing a new implicit numerical method that effectively handles these challenges in both relativistic and nonrelativistic regimes.

Contribution

It introduces a novel implicit solution method in Euler variables that overcomes hyperbolicity loss issues in cold plasma models with density-dependent collision coefficients.

Findings

The method successfully captures the loss of smoothness in solutions.

Computational results align with theoretical predictions.

The approach is applicable to both relativistic and nonrelativistic cases.

Abstract

We study a one-dimensional system of cold plasma equations taking into account electron-ion collisions in both relativistic and nonrelativistic cases. It is known that for a constant collision coefficient $\nu$, the solution to the Cauchy problem for such a system can lose smoothness. However, if the dependence of $\nu$ on the electron density $N$ is more than linear, then the solution remains globally smooth for any initial data. However, the appearance of the dependence $\nu(N)$ leads to a change in the type of the system, it loses hyperbolicity, which leads to computational problems. In this paper, we propose a new implicit solution method in Euler variables that overcomes these difficulties. It can be used in both nonrelativistic and relativistic cases and is tested for the threshold case of a linear dependence $\nu(N)=\nu_1+\nu_0 N$, when smoothness can still be lost. The computational experiments carried out are in full agreement with the available theoretical results.