A second-order accurate Super TimeStepping formulation for anisotropic thermal conduction

Astrophysical fluid dynamical problems rely on efficient numerical solution techniques for hyperbolic and parabolic terms. Efficient techniques are available for treating the hyperbolic terms. Parabolic terms, when present, can dominate the time for evaluating the solution, especially when large meshes are used. This stems from the fact that the explicit time-step for parabolic terms is proportional to the square of the mesh size and can become unusually small when the mesh is large. Multigrid-Newton–Krylov methods can help, but usually require a large number of iterations to converge. Super TimeStepping schemes are an interesting alternative, because they permit one to take very large overall time-steps for the parabolic terms while using only a modest number of explicit time-steps. Super TimeStepping schemes of the type used in astrophysics have, so far, been only first-order accurate in time and prone to instabilities.