Harmonic Initial-Boundary Evolution in General Relativity

TL;DR

This paper develops a stable, well-posed numerical algorithm for simulating gravitational fields in harmonic coordinates, demonstrating its effectiveness through various test scenarios and boundary condition strategies.

Contribution

It introduces a stable, well-posed initial-boundary evolution algorithm for general relativity in harmonic coordinates with comprehensive testing and boundary condition analysis.

Findings

The algorithm is stable and convergent in gauge wave tests.

Performance varies with different boundary conditions, including Dirichlet, Neumann, and Sommerfeld.

Constraint-preserving boundary conditions improve physical realism.

Abstract

Computational techniques which establish the stability of an evolution-boundary algorithm for a model wave equation with shift are incorporated into a well-posed version of the initial-boundary value problem for gravitational theory in harmonic coordinates. The resulting algorithm is implemented as a 3-dimensional numerical code which we demonstrate to provide stable, convergent Cauchy evolution in gauge wave and shifted gauge wave testbeds. Code performance is compared for Dirichlet, Neumann and Sommerfeld boundary conditions and for boundary conditions which explicitly incorporate constraint preservation. The results are used to assess strategies for obtaining physically realistic boundary data by means of Cauchy-characteristic matching.