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.

中文翻译：

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支持计算天体物理学和空间科学的高阶测地线网格框架的技术。

天体物理学，空间物理学和地球物理学中的许多重要问题都涉及到球形物体（例如恒星或行星）附近（可能是离子化的）气体的流动。这样的系统的几何形状自然倾向于基于球形网格的数值方案。尽管具有正交性，但由于极轴上的奇异性，极坐标（纬度-经度）网格仍然不适合计算，从而导致区域大小的高度不均匀分布。结果是：（a）由于区域纵横比的较大变化而导致的精度损失，以及（b）由于时间步长的严格限制而导致的计算效率差。测地线网格基于使用柏拉图式实体作为模板的中心投影，解决了各向异性问题，但增加了所得计算机代码的复杂性。我们在三角测地网格（TGM）上描述了Euler和MHD方程组的一种新的有限体积实现，该精度在空间和时间上精确到四阶，并且保留了磁场对机器精度的发散。本文详细讨论了TGM的生成，域分解技术，三维保守重构和时间步长。