ABSTRACT
In Section 1.3.3 we presented the quasilinear system of MHD equations for ideal and infinitely conducting plasma. It was shown to be hyperbolic, and a nondegenerate system of eigenvectors was written out. As mentioned earlier, solutions to hyperbolic systems, in general, must contain discontinuities. This, of course, is also valid for MHD systems and therefore necessitates the development of numerical methods for their solution in the conservation-law form. This system must include the integral laws of mass, momentum, and energy conservation with the action of electromagnetic forces taken into account. It must also be supplemented by the Maxwell equations describing behavior of the electromagnetic field. The latter subsystem under certain conditions can also be written in the conservationlaw form. For completeness, in Section 5.1 we are going to describe the assumptions that are usually adopted when obtaining the classical hyperbolic MHD system. In Sections 5.2 and 5.3 we outline the possible types of discontinuities intrinsic in solutions of the MHD equations and indicate those that satisfy the evolutionary property. In Section 5.4 we present various approaches to obtaining approximate solutions to the MHD Riemann problem in the one-dimensional statement. This section is devoted to numerical methods for MHD equations based, in particular, on the Riemann problem solvers. We also present the results of numerous numerical tests. In Section 5.5 we describe some peculiarities of numerical solution of MHD equations by shock-capturing methods. They are related to the fact that numerical viscosity and conductivity can never be avoided in the finite difference and finite volume methods. This sometimes results in the origin of nonevolutionary discontinuities or combinations of discontinuities. In Section 5.6 we discuss the possibility of extension of the Roe-type procedure to MHD problems involving strong background magnetic field. Section 5.7 contains some considerations on the numerical implementation of the magnetic field divergence-free condition (the absence of magnetic charge). And, finally, in Section 5.8 we present the application of numerical methods to the two-and three-dimensional problem of the solar wind interaction with the magnetized interstellar medium. We discuss in this chapter only nonrelativistic MHD. A robust Godunov-type scheme for relativistic MHD has recently been suggested by Komissarov (1999).
