Discontinuous collocation and symmetric integration methods for distributionally-sourced hyperboloidal partial differential equations

TL;DR

This paper introduces a novel numerical method for solving hyperbolic PDEs with distributional sources, such as Dirac delta functions, relevant in modeling binary black hole systems, emphasizing discontinuity handling and symmetry preservation.

Contribution

It develops a class of time-steppers that handle discontinuities and preserve time reversal symmetry, improving accuracy in simulating distributionally-sourced hyperbolic PDEs.

Findings

Effective domain-wide source conversion demonstrated

Time-reversal symmetric time-steppers developed

Numerical tests on wave equations show improved accuracy

Abstract

This work outlines a time-domain numerical integration technique for linear hyperbolic partial differential equations sourced by distributions (Dirac $\delta$-functions and their derivatives). Such problems arise when studying binary black hole systems in the extreme mass ratio limit. We demonstrate that such source terms may be converted to effective domain-wide sources when discretized, and we introduce a class of time-steppers that directly account for these discontinuities in time integration. Moreover, our time-steppers are constructed to respect time reversal symmetry, a property that has been connected to conservation of physical quantities like energy and momentum in numerical simulations. To illustrate the utility of our method, we numerically study a distributionally-sourced wave equation that shares many features with the equations governing linear perturbations to black holes sourced by a point mass.