Geodesic Mesh MHD: A New Paradigm for Computational Astrophysics and Space Physics Applied to Spherical Systems

Abstract: A majority of astrophysical systems tend to be spherical. The same is true for problems in space physics. Even so, simulations of such systems tend to be done on logically Cartesian meshes. The r-theta-phi meshes that are often used have a few major deficiencies: a) They don’t cover the sphere uniformly, b) The coordinate singularity on-axis results in a loss of accuracy, c) The timestep is seriously diminished. All these problems are associated with using an inadequate coordinate system for doing the calculation. As astrophysicists and space physicists actively take on the issues of MHD turbulence, it becomes increasingly important to have high orders of accuracy. High accuracy schemes are the only way of reducing the dissipation and dispersion that should be held down in turbulence simulations.

In this work we show that an excellent alternative is available. The most isotropic covering of the sphere comes from icosahedrally generated geodesic meshes. The elements of such a mesh are inherently curved and we show that isoparametric mapping provides an extraordinarily accurate re-mapping strategy for the sphere. We show that divergence-free MHD calculations can be done on such meshes with no loss of on-core processing efficiency or parallelism. To improve accuracy we use a combination of: a) WENO methods, b) A re-formulation to support high order divergence-free reconstruction of magnetic fields, c) Multidimensional Riemann solvers, d) Higher order timestepping to match the spatial accuracy. The resulting capability achieves provably high order of accuracy and high levels of parallelism. Several stringent test problems are presented and a few frontline applications are shown to highlight the utility of our approach.