Numerical Integration of Nonlinear Wave Equations for General Relativity

TL;DR

This paper presents a second-order numerical method for solving nonlinear wave equations in general relativity, demonstrating accuracy and convergence through tests on Gowdy T$^3$ cosmology models.

Contribution

It introduces a second-order numerical implementation for nonlinear wave equations in general relativity, utilizing the Gowdy T$^3$ cosmology as a test case.

Findings

Achieved second-order convergence in simulations.

Validated numerical results against analytical solutions.

Demonstrated flexibility in space-time slicing.

Abstract

A second-order numerical implementation is given for recently derived nonlinear wave equations for general relativity. The Gowdy T$^3$ cosmology is used as a test bed for studying the accuracy and convergence of simulations of one-dimensional nonlinear waves. The complete freedom in space-time slicing in the present formulation is exploited to compute in the Gowdy line-element. Second-order convergence is found by direct comparison of the results with either analytical solutions for polarized waves, or solutions obtained from Gowdy's reduced wave equations for the more general unpolarized waves. Some directions for extensions are discussed.