The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method

TL;DR

This paper introduces a novel numerical method for solving complex boundary value problems in nonlinear scalar fields, utilizing adapted coordinates, multipole expansions, and eigenspectral techniques, demonstrating high accuracy even with strong nonlinearities.

Contribution

The paper presents a new numerical approach combining adapted coordinates, multipole filtering, and eigenspectral methods for efficient, accurate solutions of nonlinear scalar field problems with mixed hyperbolic-elliptic character.

Findings

The method achieves high accuracy in nonlinear scalar models.

It efficiently handles boundary value problems with mixed hyperbolic and elliptic nature.

Outgoing wave approximations remain accurate under strong nonlinearities.

Abstract

The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing wave spacetimes and then extracts approximate solutions of the physical problem, with outgoing waves. The method requires solution of a boundary value problem with a mixed (hyperbolic and elliptic) character. We present here a new numerical method for such problems, based on three innovations: (i) a coordinate system adapted to the geometry of the problem, (ii) an expansion in multipole moments of these coordinates and a filtering out of higher moments, and (iii) the replacement of the continuum multipole moments with their analogs for a discrete grid. We illustrate the efficiency and accuracy of this method with nonlinear scalar model problems. Finally, we take advantage of the ability of this method to handle highly nonlinear models to demonstrate that the outgoing approximations extracted from the standing wave solutions are highly accurate even in the presence of strong nonlinearities.