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仿真工具有一系列共同的要求：它们需要高精度地鲁棒地模拟球体周围的磁流体动力学（MHD）流动。建立一个基于空间物理学和天体物理学共同需求的计算框架，将释放这两个相关研究领域之间的重要协同作用。MHD方程是流体动力学的Navier-Stokes方程和电磁学的麦克斯韦方程的组合。 因此，MHD方程需要包含流体动力学流体运动的数值求解器，并强制执行发散自由磁场，即没有磁单极子，对几何域的要求（由多边形网格近似）。MHD方程的性质将求解方法与底层网格紧密耦合，因此有必要开发新的算法，在新的网格结构上（无发散）重建磁场。天体物理和空间物理系统的仿真工具有一个共同的要求：它们需要高精度地鲁棒地模拟球体周围的磁流体动力学（MHD）流动。这个多学科的项目将开发算法，从应用数学的鲁棒性，高精度的非相对论MHD测地线网格。在过去的几年中，新的计划，模拟守恒律与真正的多维发散自由近似黎曼解的应用程序已经开发。目前，这些黎曼求解器仅适用于MHD的二维矩形结构网格。该项目将采用测地线网格，为模拟球体周围的磁流体动力学流动提供尽可能好的覆盖范围，并结合Delaunay三角测量，以实现高精度。矢量场的无发散公式可以在这些三角形网格上找到。 MHD系统被表述为守恒律系统。利用传统的守恒定律，通量可以在逐维的基础上演化。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黎曼解算器的多维逆风成为可能。这种求解器策略再次紧密耦合到网格结构。

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.