Collaborative Research: Arbitrary Order Structure-Preserving Discontinuous Galerkin Methods for Compressible Euler Equations With Self-Gravity in Astrophysical Flows

该项目旨在开发有效和高效的计算方法来模拟自引力天体物理流，包括核心塌陷超新星。该研究将对广泛的应用产生直接影响，包括天体物理学，气候建模，流体和气体动力学以及水利工程。该项目将汇集数学、天体物理学和计算科学方面的专业知识，开发新颖的数学和计算工具，以提高我们对宇宙中一些最具活力的事件进行建模的能力。研究人员将利用各自机构正在进行的研究活动，在跨学科环境中培养学生。在这个合作项目中，研究人员将设计新颖的任意阶不连续Galerkin有限元方法，用于由可压缩欧拉方程和具有自重力的理想磁流体动力学方程控制的天体物理流动。这些模型系统将适用于许多天体物理环境，包括大质量恒星的后期演化和爆炸，以及对它们的致密残余物（称为中子星）的研究。该项目将利用结构保持数值方法，即，在离散水平上保持基础物理模型的基本连续属性，例如能量守恒、渐近极限和最大值原理。该研究旨在设计这样的结构保持方法，以便在相对粗糙的网格上为模型问题及其天体物理应用提供稳定和有效的数值方法，并减轻标准数值方法的潜在问题，包括非物理和病态解的出现，数值不稳定性，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project aims to develop effective and efficient computational methods to model self-gravitating astrophysical flows, including the core-collapse supernovae. The research will have a direct impact on a wide range of applications, including astrophysics, climate modeling, fluid and gas dynamics, and hydraulic engineering. The project will bring together expertise in mathematics, astrophysics, and computational science to develop novel mathematical and computational tools that will improve our ability to model some of the most energetic events in the Universe. The investigators will leverage ongoing research activities at their respective institutions to train students in an interdisciplinary environment. In this collaborative project, the investigators will design novel arbitrary order discontinuous Galerkin finite element methods for astrophysical flows governed by the compressible Euler and ideal magnetohydrodynamics equations with self-gravity. These model systems will be applicable to many astrophysical settings, including the late-stage evolution and explosion of massive stars, and the study of their compact remnants known as neutron stars. The project will exploit structure-preserving numerical methods, i.e., preserving fundamental continuum properties of the underlying physical model, such as energy conservation, asymptotic limits, and maximum principles, at the discrete level. The research aims to design such structure-preserving methods in order to provide stable and efficient numerical methods for the model problem and its astrophysical applications on relatively coarse meshes, and to mitigate potential issues with standard numerical methods, including the appearance of non-physical and ill-conditioned solutions, numerical instability, and code breakdown.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.