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

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 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.

KEYWORDS:

• Gravitational waves
• early universe
• aeroacoustics
• turbulence

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Alberto Roper Pol http://orcid.org/0000-0003-4979-4430

Axel Brandenburg http://orcid.org/0000-0002-7304-021X

Tina Kahniashvili http://orcid.org/0000-0003-0217-9852

Arthur Kosowsky http://orcid.org/0000-0002-3734-331X

Sayan Mandal http://orcid.org/0000-0003-3129-5799

Notes

1 https://github.com/pencil-code, DOI:10.5281/zenodo.2315093

Additional information

Funding

This work was supported by National Science Foundation through the Astrophysics and Astronomy Grant Program (grants 1615100 & 1615940), the University of Colorado through the George Ellery Hale visiting faculty appointment, and the Georgian Shota Rustaveli National Science Foundation (Georgia) (grant FR/18-1462). Simulations presented in this work have been performed with computing resources provided by the Swedish National Allocations Committee at the Center for Parallel Computers at the Royal Institute of Technology in Stockholm. This work utilised the Summit supercomputer, which is supported by the National Science Foundation (award No. CNS-0821794), the University of Colorado Boulder, the University of Colorado Denver, and the National Center for Atmospheric Research.