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

FRG：协作研究：开发用于天体物理学和空间物理学的自适应测地网格 MHD 的数学算法

• 批准号: 1361197
• 负责人: Dinshaw Balsara
• 金额: $ 28.12万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361197&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Developing; Mathematical

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. 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 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. Additionally, 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.

天体物理和空间物理系统的仿真工具有一组共同的要求：它们需要以高精度稳健地模拟球体周围的磁流体动力学 (MHD) 流。这个多学科项目将开发应用数学算法，以实现测地网格上稳健、高精度的非相对论 MHD。在过去的几年中，已经开发出用于模拟守恒定律的新方案，该方案具有真正的多维无散度近似黎曼求解器的应用。目前，这些黎曼求解器仅适用于 MHD 的二维矩形结构网格。该项目将采用测地网格为球体周围磁流体动力流的模拟提供尽可能最佳的覆盖范围，并结合德劳内三角测量以实现高精度。可以在这些三角形网格上找到矢量场的无散公式。天体物理和空间物理系统的仿真工具有一组共同的要求：它们需要以高精度稳健地模拟球体周围的磁流体动力学 (MHD) 流。基于空间物理学和天体物理学的共同需求构建计算框架，将释放这两个相关研究领域之间的重要协同作用。 MHD 方程是流体动力学纳维-斯托克斯方程和电磁学麦克斯韦方程的组合。 因此，MHD 方程需要数值求解器，其中包含流体动力流体运动并强制无发散磁场，即无磁单极子，对由多边形网格近似的几何域的要求。 MHD 方程的本质将求解方法与底层网格紧密结合在一起，因此有必要开发新的算法来在新颖的网格结构上进行无发散的磁场重建。此外，MHD 系统被制定为守恒定律系统。利用传统的守恒定律，通量可以在逐维的基础上演化。不同通量分量在对合约束系统中耦合的事实也为基于多维黎曼求解器的多维逆风提供了理由。这种求解器策略再次与网格结构紧密耦合。