Nonlinear Magnetohydrodynamics: Numerical Concepts

The ideal MagnetoHydroDynamic (MHD) equations accurately describe the macroscopic dynamics of a perfectly conducting plasma. Adopting a continuum, single fluid description in terms of the plasma density , velocity v, thermal pressure p and magnetic field B, the ideal MHD system expresses conservation of mass, momentum, energy, and magnetic flux. This nonlinear, conservative system of 8 partial differential equations enriches the Euler equations governing the dynamics of a compressible gas with the dynamical influence through the Lorentz force - and evolution - through the additional induction equation - of the magnetic field B. In multi-dimensional problems, the topological constraint expressed by the Maxwell equation (nabla) B = 0, represents an additional complication for numerical MHD. Basic concepts of shock-capturing high-resolution schemes for computational MHD are presented, with an emphasis on how they cope with the thight physical demands resulting from nonlinearity, compressibility, conservation, and solenoidality.