Exact boundary conditions in numerical relativity using multiple grids: scalar field tests

TL;DR

This paper introduces a fourth order accurate finite difference scheme for Cauchy-Characteristic Matching in numerical relativity, enabling precise boundary conditions and gravitational wave extraction near black holes, demonstrated through scalar field tests.

Contribution

It presents a novel fourth order finite difference CCM scheme for wave equations around black holes, improving accuracy and boundary condition handling in numerical relativity.

Findings

The scheme is demonstrated to be fourth order convergent.

It successfully reproduces late-time tail decay of scalar fields.

Numerical experiments validate the method's effectiveness.

Abstract

Cauchy-Characteristic Matching (CCM), the combination of a central 3+1 Cauchy code with an exterior characteristic code connected across a time-like interface, is a promising technique for the generation and extraction of gravitational waves. While it provides a tool for the exact specification of boundary conditions for the Cauchy evolution, it also allows to follow gravitational radiation all the way to infinity, where it is unambiguously defined. We present a new fourth order accurate finite difference CCM scheme for a first order reduction of the wave equation around a Schwarzschild black hole in axisymmetry. The matching at the interface between the Cauchy and the characteristic regions is done by transfering appropriate characteristic/null variables. Numerical experiments indicate that the algorithm is fourth order convergent. As an application we reproduce the expected late-time tail decay for the scalar field.