Solution of hyperbolic system with magnetic field

Abstract

This paper is concerned with the solution of the Riemann problem (RP) for the hyperbolic system with the logarithmic equation of state and magnetic field. The formation of vacuum states and delta shock wave as magnetic field and pressure vanish have been discussed. First, the Riemann problem for the magnetogasdynamic system is solved. Further, we have constructed the solutions for the pressureless and vanishing magnetic field system (i.e. transport equations). In the context of vacuum states and delta shocks, it is shown that the solution of RP consisting two shocks converges to the delta shock wave (δ-shock) solution of transport equations, and the Riemann solutions consisting two rarefaction waves converge to the vacuum state solution (i.e. intermediate state between two-contact discontinuity solution of transport equations). Hence, it is proved that the solutions of system with logarithmic equation of state and magnetic field converge to the solution of corresponding system as magnetic field and pressure vanish, which shows that the solution of the RP for the hyperbolic system with the logarithmic equation of state and magnetic field is stable.