The compressible magnetohydrodynamics (MHD) equations are a nonlinear system of partial differential equations (PDEs) that describe electrically conducting fluids. We will give an overview of the mathematical properties of the MHD equations and will discuss numerical methods that can be used to solve them, illustrated by applications in space plasma physics.
We will first analyze the properties of MHD as a system of nonlinear hyperbolic conservation laws. In terms of mathematical structure, MHD is in many ways similar to the compressible Euler equations of gas dynamics, but as a hyperbolic system MHD possesses a richer structure than the Euler equations and we will highlight these differences. For example, MHD features three families of nonlinear waves that are anisotropic, and it allows for several different types of shock waves, including overcompressive shocks and compound shocks, which have interesting stability properties. This rich hyperbolic structure leads to surprisingly intricate flow phenomena even in basic problem configurations, for example in the topology of MHD bow shock flows.
Next we will give an overview of numerical methods that can be used to solve the MHD equations. Since MHD is a system of nonlinear hyperbolic conservation laws similar to Euler, standard finite volume methods from gas dynamics are the natural starting point. They can be applied without much change, except that special care has to be taken to handle the divergence-free constraint of the MHD magnetic field. We will give an overview of the different techniques that have been developed to handle this difficulty, including projection, an eight-wave formulation, constrained transport, and generalized Lagrange multipliers. We will also discuss recent work on multi-dimensional fourth-order accurate finite volume methods for MHD, and adaptive cubed-sphere grids for space physics problems in spherical domains.
Finally we will discuss efficient numerical methods for computing stationary compressible fluid flows in which transitions from subsonic to supersonic flow occur. Rather than employing time marching methods, we solve the stationary equations directly. We first present a solution method for one-dimensional flows with critical points. The method is based on a dynamical systems formulation and uses adaptive integration combined with nonlinear shooting. The method can be extended easily to handle flows with shocks, using a Newton method applied to the Rankine-Hugoniot relations. Extension to flows with heat conduction is also discussed. The method is illustrated for hydrogen escape from extrasolar planets, quasi one-dimensional nozzle flow and black hole accretion. The presentation will conclude with some thoughts on how the approach presented can be generalized to problems in higher dimensions.