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Many important problems in astrophysics, space physics, and geophysicsinvolve flows of (possibly ionized) gases in the vicinity of a sphericalobject, such as a star or planet. The geometry of such a system naturallyfavors numerical schemes based on a spherical mesh. Despite its orthogonalityproperty, the polar (latitude-longitude) mesh is ill suited for computationbecause of the singularity on the polar axis, leading to a highly non-uniformdistribution of zone sizes. The consequences are (a) loss of accuracy due tolarge variations in zone aspect ratios, and (b) poor computational efficiencyfrom a severe limitations on the time stepping. Geodesic meshes, based on acentral projection using a Platonic solid as a template, solve the anisotropyproblem, but increase the complexity of the resulting computer code. Wedescribe a new finite volume implementation of Euler and MHD systems ofequations on a triangular geodesic mesh (TGM) that is accurate up to fourthorder in space and time and conserves the divergence of magnetic field tomachine precision. The paper discusses in detail the generation of a TGM, thedomain decomposition techniques, three-dimensional conservative reconstruction,and time stepping.

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences