FRG: Collaborative Research: Developing Mathematical Algorithms for Adaptive, Geodesic Mesh MHD for use in Astrophysics and Space Physics

天体物理和空间物理系统的模拟工具有一系列共同的要求：它们需要高精度地稳健地模拟球形物体周围的磁流体流。建立一个基于空间物理和天体物理共同需求的计算框架，将在这两个联合研究领域之间释放出重要的协同效应。MHD方程是流体力学的Navier-Stokes方程和电磁的Maxwell方程的组合。因此，MHD方程需要包含流体动力运动的数值求解器，并在几何区域(由多边形网格近似)上实施无散度磁场，即没有磁单极子的要求。MHD方程的性质将求解方法紧密地耦合到底层网格上，这使得有必要开发新的算法来重建新型网格结构上的磁场(无散度)。用于天体物理和空间物理系统的模拟工具有一组共同的要求：它们需要高精度地稳健地模拟球形物体周围的磁流体力学(MHD)流动。这个多学科项目将从应用数学出发，在测地线网格上开发健壮、高精度的非相对论MHD算法。在过去的几年里，已经发展了新的方案来模拟守恒律，并使用真正的多维无散度近似黎曼求解器来应用。目前，这些黎曼解算器仅适用于MHD的二维矩形结构网格。该项目将采用测地线网格，为模拟球形物体周围的磁流体流提供尽可能好的覆盖范围，并采用Delaunay三角剖分以实现高精度。在这些三角网格上可以找到矢量场的无散度公式。MHD系统是作为一种守恒定律系统而制定的。根据传统的守恒定律，通量可以在一个维度一个维度的基础上演化。不同的通量分量在对合约束系统中耦合的事实也使得基于多维Riemann求解器的多维迎风成为可能。这样的求解器策略再次紧密地耦合到网格结构。

Simulation tools for astrophysical and space physics systems share a set of common requirements: they need to robustly simulate magnetohydrodynamic (MHD) flows around spherical bodies with high accuracy. Building a computational framework, based on shared needs in space physics and astrophysics, will unleash important synergies between these two allied fields of study. The MHD equations are a combination of the Navier-Stokes equations for fluid dynamics and Maxwell's equations for electromagnetism. Thus, the MHD equations require numerical solvers that incorporate the hydrodynamic fluid motion and enforce the divergence free magnetic field, i.e. no magnetic monopoles, requirements on the geometric domain (which is approximated by a polygonal mesh). The nature of the MHD equations closely couples solution methodologies to the underlying mesh, making it necessary to develop new algorithms for the (divergence-free) reconstruction of the magnetic field on novel mesh structures.Simulation tools for astrophysical and space physics systems share a set of common requirements: they need to robustly simulate magnetohydrodynamic (MHD) flows around spherical bodies with high accuracy. This multidisciplinary project will develop algorithms from applied mathematics for robust, highly accurate non-relativistic MHD on geodesic meshes. In the past few years new schemes for simulating conservation laws with truly multi-dimensional divergence free approximate Riemann solvers for applications have been developed. Currently, these Riemann solvers are only available for two-dimensional rectangular structured meshes for MHD. This project will employ a geodesic mesh to provide the best possible coverage for simulations of magnetohydrodynamic flows around spherical bodies and to incorporate Delaunay triangulation to achieve high accuracy. Divergence-free formulations of vector fields can be found on these triangular meshes. The MHD system is formulated as a system of conservation laws. With a traditional conservation law, the fluxes can be evolved on a dimension-by-dimension basis. The fact that different flux components are coupled in an involution-constrained system also makes a case for multidimensional upwinding based on multidimensional Riemann solvers. Such solver strategies are again intimately coupled to the mesh structure.