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

TL;DR

This paper reviews the mathematical theory of continuum and discrete initial-boundary value problems for hyperbolic PDEs, focusing on applications to numerical relativity and Einstein's equations, including well-posed formulations and high-order numerical methods.

Contribution

It provides a comprehensive overview of the theory and numerical techniques for solving Einstein's equations on finite domains, emphasizing well-posed formulations and high-order methods.

Findings

Well-posed initial and boundary value formulations for Einstein's equations

Application of multi-domain high-order finite difference methods

Discussion of spectral methods for numerical relativity

Abstract

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.