Modern Numerical Schemes for Solving Magnetohydrodynamic Equations

In this paper several modern shock-capturing schemes for solving hydrodynamic and magnetohydrodynamic equations are reviewed. This review covers a wide range of explicit and implicit schemes as well as those in which adaptive mesh refinement methods are adopted. As these numerical schemes are based on Riemann solvers which use Godunov-type techniques, they are well suited for strong shocks and other discontinuities without oscillations in the flow variables. Some other numerical issues as grid generation, divergence cleaning, and an application of MHD schemes to several problems in coronal and interplanetary physics are discussed.