On the numerical solution of one extended hyperbolic system

Numerical simulation of the influence of an external constant magnetic field on plane relativistic plasma oscillations is carried out. For this purpose, an algorithm is constructed in Lagrangian variables based on an extended system of hyperbolic equations. An important property of the numerical method is the dependence of its accuracy only on the smoothness properties of the solution. In addition, control over the intersection of electronic trajectories is used to fix the moment of breaking of oscillations. Sufficient conditions for the existence and non-existence of a smooth solution of the problem in the first period are analytically obtained. It was found out that the external magnetic field cannot prevent the breaking of oscillations in principle, even for the case of an arbitrarily small initial deviation from the equilibrium position. Numerical experiments clearly illustrate the relativistic breaking of the upper hybrid oscillations. It is shown that an external magnetic field can both accelerate and slow down the breaking process depending on the choice of the initial condition for the transverse component of the electron pulse.