Multidomain Spectral Method for the Helically Reduced Wave Equation

TL;DR

This paper introduces a spectral domain decomposition method for solving the helically reduced wave equation, including nonlinear models, aiming for high accuracy in simulating binary inspiral phenomena.

Contribution

It presents a novel spectral domain decomposition approach for the linear and nonlinear helically reduced wave equation, enhancing accuracy and flexibility in binary inspiral simulations.

Findings

Effective spectral domain decomposition method developed

Achieved high accuracy in solving linear HRWE

Extended approach to nonlinear scalar models

Abstract

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the "eigenspectral method." Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.