Abstract
Many important problems in astrophysics, space physics, and geophysics involve flows of (possibly ionized) gases in the vicinity of a spherical object, such as a star or planet. The geometry of such a system naturally favors numerical schemes based on a spherical mesh. Despite its orthogonality property, the polar (latitude-longitude) mesh is ill suited for computation because of the singularity on the polar axis, leading to a highly non-uniform distribution of zone sizes. The consequences are (a) loss of accuracy due to large variations in zone aspect ratios, and (b) poor computational efficiency from a severe limitations on the time stepping. Geodesic meshes, based on a central projection using a Platonic solid as a template, solve the anisotropy problem, but increase the complexity of the resulting computer code. We describe a new finite volume implementation of Euler and MHD systems of equations on a triangular geodesic mesh (TGM) that is accurate up to fourth order in space and time and conserves the divergence of magnetic field to machine precision. The paper discusses in detail the generation of a TGM, the domain decomposition techniques, three-dimensional conservative reconstruction, and time stepping.
Highlights
Objects in the universe tend to assume a spherical shape owing to the central nature of the gravitational force
Construction of a triangular geodesic mesh (TGM) begins with inscribing an icosahedron inside a sphere and centrally projecting its edges to the surface of the sphere, see the top row of Fig. 1
We describe the five separate constraints imposed on the magnetic field modes that ensure that the magnetic field remains divergence-free in the integral sense, and functionally at any location within the zone
Summary
Objects in the universe tend to assume a spherical shape owing to the central nature of the gravitational force. Construction of a TGM begins with inscribing an icosahedron inside a sphere (in the rest of this paper we will always assume that the sphere has a unit radius, unless stated otherwise) and centrally projecting its edges to the surface of the sphere, see the top row of Fig. 1. This projection generates a division 0 tesselation that includes 12 vertices, 20 triangular faces, called t-faces and 30 edges, called t-edges (these names are chosen to distinguish them from the faces and edges oriented in the radial direction produced by the radial extrusion of the mesh that bear the prefix “r”).
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