Numerical study of loss of hyperbolicity using a cold plasma model

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.