连续统与离散初边值问题与爱因斯坦场方程

IF 26.3 2区 物理与天体物理 Q1 PHYSICS, PARTICLES & FIELDS
Olivier Sarbach, Manuel Tiglio
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引用次数: 108

摘要

物理学中的许多演化问题都是用无限域上的偏微分方程来描述的;因此,对于给定的初始数据集,人们对这些问题的解决方案感兴趣。一个突出的例子是爱因斯坦引力理论中的双黑洞问题,在这个问题中,人们计算了两个黑洞的灵感,合并和环落发出的引力辐射。强大的数学工具可以用来建立关于解的定性陈述,例如它们的存在性,唯一性,对初始数据的连续依赖性,或者它们在大时间尺度上的渐近行为。然而,人们通常对计算解本身感兴趣,除非偏微分方程非常简单,或者初始数据具有高度对称性,否则这种计算需要通过数值离散化进行近似。当在机器上解决这种离散问题时,人们面临着计算资源有限的限制,这导致用有限的计算机网格代替无限的连续域。这又导致了一个离散的初边值问题。我们希望能够以高精度恢复网格间距收敛于零并将边界推至无穷大的极限的精确解。本文的目的是回顾一些必要的理论,以理解由双曲型偏微分方程引起的连续统和离散初始边值问题,并讨论其在数值相对论中的应用;特别地,我们给出了爱因斯坦方程的适定初值和初边值公式,并讨论了求解它们的多域高阶有限差分和谱方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.

The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

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来源期刊
Living Reviews in Relativity
Living Reviews in Relativity 物理-物理:粒子与场物理
CiteScore
69.90
自引率
0.70%
发文量
0
审稿时长
20 weeks
期刊介绍: Living Reviews in Relativity is a peer-reviewed, platinum open-access journal that publishes reviews of research across all areas of relativity. Directed towards the scientific community at or above the graduate-student level, articles are solicited from leading authorities and provide critical assessments of current research. They offer annotated insights into key literature and describe available resources, maintaining an up-to-date suite of high-quality reviews, thus embodying the "living" aspect of the journal's title. Serving as a valuable tool for the scientific community, Living Reviews in Relativity is often the first stop for researchers seeking information on current work in relativity. Written by experts, the reviews cite, explain, and assess the most relevant resources in a given field, evaluating existing work and suggesting areas for further research. Attracting readers from the entire relativity community, the journal is useful for graduate students conducting literature surveys, researchers seeking the latest results in unfamiliar fields, and lecturers in need of information and visual materials for presentations at all levels.
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