  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.

Computational Physics;Instrumentation and Methods for Astrophysics