Initial-boundary Value Formulations Research Articles

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.

A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations

We discuss the initial-boundary value problem of general relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, we show that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar Ψ0 yields a priori estimates on the metric, connection and curvature fields. These estimates are expected to be useful for obtaining a well-posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity. To present a simple and explicit example, we consider the 3 + 1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse, an a priori specified shift vector and our boundary conditions lead to a well-posed problem. The method discussed here is quite general and should also yield well-posed problems for different ways of writing the evolution equations, including first-order symmetric hyperbolic or mixed first-order second-order formulations. Well-posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere.

A MODEL PROBLEM FOR THE INITIAL-BOUNDARY VALUE FORMULATION OF EINSTEIN'S FIELD EQUATIONS

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of space–time with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein–Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein–Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background.

Detecting ill posed boundary conditions in general relativity

A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many-parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace–Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: (i) boundary conditions that have been shown to preserve the constraints, (ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero, and (iii) boundary conditions that result from considering the projection of Einstein’s equations along the normal to the boundary surface. While we show that in case (i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases.

Well Posed Constraint-Preserving Boundary Conditions for the Linearized Einstein Equations

In this paper we address the problem of specifying boundary conditions for Einstein's equations when linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. The boundary conditions we work out guarantee that the constraints are satisfied provided they are satisfied on the initial slice and ensures a well posed initial-boundary value formulation. We consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case.