IMEX evolution of scalar fields on curved backgrounds

TL;DR

This paper introduces an implicit-explicit evolution strategy using spectral methods to efficiently simulate scalar fields on curved backgrounds, addressing challenges in long-term binary black hole simulations.

Contribution

It develops a novel implicit-explicit approach combined with spectral methods to overcome stability constraints in simulating scalar fields on curved spacetimes.

Findings

Successful implementation of the method on Schwarzschild backgrounds

Demonstrated stability and efficiency in numerical experiments

Addressed issues relevant to binary black hole evolutions

Abstract

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.