Magnetohydrodynamics (MHD) is arguably the most popular mathematical model for the macroscopic simulations of fusion plasmas. In this talk we will focus on the resistive single-fluid MHD equations, the solutions of which can exhibit near-singular layers (or even discontinuities in the absence of diffusion terms). We rely on locally adaptive structured mesh refinement (AMR) methods to mitigate the separation of spatial scales in MHD. We will present results from AMR simulations of MHD applications relevant to the fusion program. These will include pellet injection, a proven method to refuel tokamaks; magnetic reconnection which is a canonical problem in plasma physics involving thin current sheets; and an example in MHD shock refraction where five or more discontinuities meet at a single point.

For a tokamak fusion plasma, the presence of a large background field and toroidal geometry results in a large separation of temporal scales. Explicit time-stepping methods to simulate fusion plasmas become prohibitively expensive due to the CFL constraint on the time-step. To overcome the temporal stiffness associated with the fast compressive and Alfven waves in MHD, we have developed a nonlinearly implicit time stepping method using a Jacobian-Free Newton-Krylov approach (JFNK) and begun exploring nonlinear multigrid methods. At the heart of our JFNK method is a PDE-operator based preconditioner (exact for a 1D system of hyperbolic PDEs), to effectively solve the resulting large ill-conditioned linear system.