Michael Pernice, Bobby Philip, Luis Chacon

Towards Implicit Resistive Magnetohydrodynamics with Local Mesh Refinement

Efficient and accurate solution of the resistive magnetohydrodynamics model presents a significant challenge due to the presence of multiple length and time scales. Explicit methods introduce stringent CFL restrictions in order to maintain stability. Semi-implicit approaches relieve this restriction, but accuracy considerations limit time steps to values that are much smaller than the time scale of phenomena of interest. Recently, the practicality of fully implicit approaches that use second-order time differencing schemes and time steps that are much closer to time scales of interest has been demonstrated. Solutions so obtained agree with those obtained with an explicit scheme, but at a much lower cost. The key to the success of this approach is the combination of a Newton-Krylov method with a physics-based preconditioner that captures the key couplings of interest, together with efficient multigrid methods to implement the preconditioner.

On the other hand, it is impractical to resolve fine spatial features with a globally fine mesh, which leads to consideration of local mesh refinement. When local mesh refinement is combined with explicit or semi-implicit time stepping methods, new strategies, such as local time stepping, must be introduced. Depending on how much of the computational domain is covered by the finest mesh, the use of smaller time steps on the finest regions may dominate the cost of the calculation. Implicit methods allow a single global time step that is restricted only by the desired accuracy. The same physics-based preconditioner can still be used; however its implementation must now account for local refinement in order to remain competetive.

We describe our approach to combining implicit time stepping with local mesh refinement in a resistive magnetohydrodynamics model. The Newton-Krylov method and physics-based preconditioning remain at the heart of the scheme, but components of the preconditioner are now based on the Fast Adaptive Composite grid (FAC) method to explicitly account for local mesh refinement. Straightforward modifications to the discretization scheme that are needed at changes in mesh resolution will also be described. The parallel implementation uses SAMRAI to manage complex data structures associated with dynamic, locally refined meshes; solvers from PETSc to drive the solution process; and components built from Hypre's structured multigrid solvers to solve subproblems on the coarsest level. These various packages are coupled through a seamless interface that features the use of multiple data representations that allow the most natural implementation of different phases of the calculation while minimizing data copying.