Discontinuous collocation methods and gravitational self-force applications

TL;DR

This paper introduces a novel discontinuous collocation method that leverages a-priori information about discontinuities to accurately solve PDEs in black hole perturbation theory, improving simulations of gravitational self-force effects.

Contribution

The authors develop a high-order accurate, discontinuous collocation approach that effectively handles distributionally sourced PDEs in gravitational self-force calculations, enhancing numerical stability and efficiency.

Findings

Achieves high-order accuracy near discontinuities

Efficiently handles moving singularities in PDEs

Suitable for time-domain gravitational wave simulations

Abstract

Numerical simulations of extereme mass ratio inspirals, the mostimportant sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.