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.

中文翻译：

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连续和离散初始边值问题和爱因斯坦的场方程。

物理学中的许多演化问题都由无限域上的偏微分方程描述。因此，对于给定初始数据集的此类问题的解决方案感兴趣。一个著名的例子是爱因斯坦引力理论中的二元黑洞问题，其中计算从两个黑洞（合并和衰荡）的吸气发出的引力辐射。强大的数学工具可用于建立有关解决方案的定性说明，例如其存在性，唯一性，对初始数据的连续依赖性或其在较大时间范围内的渐近行为。但是，人们通常会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，该计算需要通过数值离散化来近似。当解决一台机器上的离散问题时，人们面临着对计算资源的有限限制，这导致用有限的计算机网格代替无限连续域。反过来，这导致离散的初始边界值问题。希望能够以高精度恢复在网格间距收敛到零且边界被推向无穷大的极限中的精确解。本文的目的是回顾一些了解连续体和离散的必要理论。由双曲型偏微分方程引起的初始边值问题，并讨论其在数值相对论中的应用；特别是，