The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence

TL;DR

This paper investigates the numerical challenges in solving gravitational wave equations sourced by hydromagnetic turbulence, revealing timestep-dependent errors and proposing an exact solution method to improve computational efficiency.

Contribution

It identifies the timestep constraint issue in gravitational wave solvers and introduces an exact solution approach to mitigate numerical degradation and reduce computational costs.

Findings

Numerical degradation of GW amplitude scales with the cube of the timestep.

Degradation affects high wavenumbers more significantly.

Using the exact solution allows for larger timesteps and reduces computational cost.

Abstract

Hydromagnetic turbulence produced during phase transitions in the early universe can be a powerful source of stochastic gravitational waves (GWs). GWs can be modelled by the linearised spatial part of the Einstein equations sourced by the Reynolds and Maxwell stresses. We have implemented two different GW solvers into the {\sc Pencil Code} -- a code which uses a third order timestep and sixth order finite differences. Using direct numerical integration of the GW equations, we study the appearance of a numerical degradation of the GW amplitude at the highest wavenumbers, which depends on the length of the timestep -- even when the Courant--Friedrichs--Lewy condition is ten times below the stability limit. This degradation leads to a numerical error, which is found to scale with the third power of the timestep. A similar degradation is not seen in the magnetic and velocity fields. To mitigate numerical degradation effects, we alternatively use the exact solution of the GW equations under the assumption that the source is constant between subsequent timesteps. This allows us to use a much longer timestep, which cuts the computational cost by a factor of about ten.