Shock-Capturing Approach and Nonevolutionary Solutions in Magnetohydrodynamics

Abstract

Shock-capturing methods have become an effective tool for the solution of hyperbolic partial differential equations. Both upwind and symmetric TVD schemes in the framework of the shock-capturing approach are thoroughly investigated and applied with great success to a number of complicated multidimensional gasdynamic problems. The extension of these schemes to magnetohydrodynamic (MHD) equations is not a simple task. First, the exact solution of the MHD Riemann problem is too multivariant to be used in regular calculations. On the other hand, the extensions of Roe's approximate Riemann problem solvers for MHD equations in general case are nonunique and need further investigation. That is why, some simplified approaches should be constructed. In this work, the second order of accuracy in time and space high-resolution Lax–Friedrichs type scheme is suggested that gives a drastic simplification of the numerical algorithm comparing to the precise characteristic splitting of Jacobian matrices. The necessity is shown to solve the full set of MHD equations for modeling of multishocked flows, even when the problem is axisymmetric, to obtain evolutionary solutions. For the numerical example, the MHD Riemann problem is used with the initial data consisting of two constant states lying to the right and to the left from the centerline of the computational domain. If the problem is solved as purely coplanar, a slow compound wave appears in the self-similar solution obtained by any shock-capturing scheme. If the full set of MHD equations is used and a small uniform tangential disturbance is added to the magnetic field vector, a rotational jump splits from the compound wave, and it degrades into a slow shock. The reconstruction process of the nonevolutionary compound wave into evolutionary shocks is investigated. Presented results should be taken into account in the development of shock-capturing methods for MHD flows.