Numerical Methods for Geometric Partial Differential Equations with Applications in Numerical Relativity

几何偏微分方程的数值方法及其在数值相对论中的应用

• 批准号: 2012857
• 负责人: Michael Holst
• 金额: $ 45万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012857&HistoricalAwards=false
• 关键词: Numerical; Methods; Geometric; Partial; Differential

This project is concerned with the approximate solution of systems of stationary and evolution partial differential equations (PDE) arising at the intersection of mathematical physics and geometric analysis. Such systems of equations, known as Geometric PDE, with both constraints and extra degrees of freedom, appear in a wide range of physical and mathematical problems; examples include Maxwell's equations (or more generally the Yang-Mills equations on a curved background), and Einstein's field equations and other Hamiltonian systems. The initial-value formulation for such systems yields a constrained evolution system which has to be augmented with side conditions in order to get a unique evolution. The non-dynamical geometric PDE (as constraints or otherwise) are of great interest in their own right; examples include the Yamabe problem, the Hamiltonian and momentum constraints in the Einstein equations, and the Monge-Ampere equations, among others. One of the most challenging features of this class of problems, for both mathematical analysis and computational simulation, is the underlying spatial domain which has the structure of a manifold with potentially complicated topology. Moreover, both the geometry and the topology may evolve over time, depending on the particular model. The results of this project have the potential for broad impact on areas of mathematics such as geometric analysis, as well as in astrophysics and general relativity. The methods developed here will contribute to the advancement of numerical methods for complex three-dimensional constrained nonlinear dynamical simulations. The simulation technology produced will provide powerful tools for the exploration of mathematical and computational models in astrophysics and relativity, as well as in some areas of pure mathematics such as geometric analysis. This project provides research training opportunities for graduate students. The primary technical aims of this project are to develop new discretization techniques for a class of geometric PDE that includes the Einstein equations. The emphasis is on modeling cases that present particular challenges for current state-of-the-art methods and software currently used for the Einstein equations, such as the case of extreme mass ration binary black hole systems. The tools will be the development of approximation theory, together with reliable and provably convergent adaptive methods, for the intrinsic discretization of the class of nonlinear geometric PDE on Riemannian 2- and 3- manifolds. Most of the approaches to date, such as surface finite element methods for two-dimensional problems, are based on exploiting the embedding of the surface into three space, and then on use of method-of-lines discretization for separating the space and time discetizations. For applications such as general relativity, a more general approach is needed that does not rely on the existence of such an embedding, and does not on an a priori spatial slicing. This project studies the development of truly intrinsic discretizations that use no extrinsic information to produce a discretization, to allow for the development of numerical methods for evolution PDE on Riemannian 2- and 3-manifolds with arbitrary topology and without imposing an a priori discrete spatial slicing. The approach is to develop atlas-based discretization techniques and space-time discretizations based on explicit tent-pitching methods or fully implicit space-time discetizations. For the design of such methods and their analysis, researchers will exploit variational crimes frameworks developed by their team and collaborators for analyzing numerical methods posed on surfaces, and through use of the finite element exterior calculus framework.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.

该项目与在数学物理学和几何分析的交集中产生的固定和进化部分微分方程（PDE）系统的近似解决方案有关。 这种方程式（称为几何PDE）具有约束和额外的自由度，出现在各种物理和数学问题中。示例包括麦克斯韦的方程（或更一般而言的弯曲背景上的杨麦克米尔斯方程），以及爱因斯坦的田间方程和其他哈密顿系统。此类系统的初始值配方产生了一个约束的演化系统，必须在侧面条件下增强，以获得独特的演化。 非动力的几何PDE（作为约束或其他方式）本身就引起了极大的兴趣；例如，爱因斯坦方程中的哈密顿和动量限制以及蒙格 - 安培方程等等。 对于数学分析和计算模拟，这类问题的最具挑战性的特征之一是基本的空间域，其具有带有潜在复杂拓扑的多种形式的结构。 此外，根据特定模型，几何形状和拓扑都可以随着时间的推移而发展。 该项目的结果可能会对数学领域（例如几何分析）以及天体物理学和总体相对论产生广泛影响。 这里开发的方法将有助于进步数值方法，以进行复杂的三维约束非线性动力学模拟。 所产生的仿真技术将为天体物理学和相对论中的数学和计算模型以及在诸如几何分析之类的纯数学领域（例如几何分析）提供强大的工具。该项目为研究生提供了研究培训机会。该项目的主要技术目的是为包括爱因斯坦方程的一类几何PDE开发新的离散技术。 重点是建模案例，这些案例对当前用于爱因斯坦方程的最新方法和软件提出了特殊的挑战，例如极端的质量分配二进制二进制黑洞系统。 这些工具将是近似理论的发展，以及可靠且可行的自适应方法，用于在Riemannian 2和3歧管上的非线性几何PDE类的内在离散化。 迄今为止，大多数方法（例如用于二维问题的表面有限元方法）是基于将表面嵌入到三个空间中，然后使用嵌入方式，然后使用离散方法来分开空间和时间识别。 对于诸如一般相对论之类的应用，需要一种更通用的方法，该方法不依赖于这种嵌入的存在，也不依赖于先验的空间切片。 该项目研究了真正固有的离散化的发展，这些离散化不使用外部信息来产生离散化，以开发用于带有任意拓扑的Riemannian 2和3个manifolds进化PDE的数值方法，而不会施加先验的离散空间切片。 该方法是基于明确的帐篷方法或完全隐式的时空识别来开发基于ATLA的离散技术和时空离散化。 为了设计此类方法及其分析，研究人员将利用其团队和合作者开发的变异犯罪框架来分析在表面上提出的数值方法，并使用有限元元素外部计算框架。该奖项反映了NSF的法定任务，并通过评估了基金会的智力效果，并通过评估了基金会的范围和广泛的范围。

项目成果

Multiplication in Sobolev spaces, revisited

• DOI: 10.4310/arkiv.2021.v59.n2.a2
• 发表时间: 2015-12
• 期刊: Arkiv för Matematik
• 作者: A. Behzadan;Michael Holst

Improved spectral representations of neutron-star equations of state

改进的中子星状态方程的谱表示

• DOI: 10.1103/physrevd.105.063031
• 发表时间: 2022
• 期刊: Physical Review D
• 影响因子: 5
• 作者: Lindblom, Lee

Local finite element approximation of Sobolev differential forms

Sobolev 微分形式的局部有限元近似

• DOI: 10.1051/m2an/2021034
• 发表时间: 2021
• 期刊: ESAIM: Mathematical Modelling and Numerical Analysis
• 作者: Gawlik, Evan;Holst, Michael J.;Licht, Martin W.
• 通讯作者: Licht, Martin W.

First Constraining Upper Limits on Gravitational-wave Emission from NS 1987A in SNR 1987A

• DOI: 10.3847/2041-8213/ac84dc
• 发表时间: 2022-06
• 期刊: The Astrophysical Journal Letters
• 作者: B. Owen;L. Lindblom;Luciano Soares Pinheiro

Symmetry Breaking and the Generation of Spin Ordered Magnetic States in Density Functional Theory Due to Dirac Exchange for a Hydrogen Molecule

• DOI: 10.1007/s00332-022-09845-2
• 发表时间: 2022-09
• 期刊: Journal of Nonlinear Science
• 影响因子: 3
• 作者: M. Holst;Houdong Hu;Jianfeng Lu;J. Marzuola;D. Song;J. Weare