Numerical study of loss of hyperbolicity using a cold plasma model

TL;DR

We address loss of hyperbolicity in a one-dimensional cold plasma model with electron-ion collisions, showing that a linear density dependence $ν(N)=ν_0 N+ν_1$ induces a Jordan block and gradient catastrophes. It proposes a robust implicit MacCormack-type scheme in Euler variables to solve the nonhyperbolic system, with predictor-corrector steps that handle the Jordan-block flux while remaining applicable to both relativistic and nonrelativistic regimes. Numerical experiments demonstrate that linear $ν(N)=ν_0 N+ν_1$ damps oscillations and can delay or prevent breaking in both regimes, with a relativistic threshold condition $ν_1 heta^{(0)}_{wb} le 0.422$ for a gradient catastrophe to appear, and with $ν_0>0$ further suppressing breaking. The results corroborate theoretical predictions and provide a practical tool for modeling laser-plasma interactions and for developing numerical methods for singularly perturbed hyperbolic systems.

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 $ν$, the solution to the Cauchy problem for such a system can lose smoothness. However, if the dependence of $ν$ 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 $ν(N)$ 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 $ν(N)=ν_1+ν_0 N$, when smoothness can still be lost. The computational experiments carried out are in full agreement with the available theoretical results.

Figures (6)

• Figure 1: Dynamics of density in nonrelativistic oscillations for $\alpha=0.4761$: the breaking effect is absent for different collision coefficients
• Figure 2: Dynamics of density in nonrelativistic oscillations for $\alpha=0.51$: the breaking effect is absent only for non-zero collision coefficients
• Figure 3: Dynamics of density in relativistic collisionless plasma: maximum over the region (black) and at the origin (red)
• Figure 4: Momentum and electric field at the moment of breaking in a relativistic collisionless plasma
• Figure 5: Dynamics of density in relativistic oscillations taking into account collisions ($\nu_0=0, \, \nu_1 \theta^{(0)}_{wv}=0.3$): maximum over the region (black) and at the origin (red).