Pseudospectral vs finite differences methods in Numerical Relativity

Abstract

Inspired by the recent discovery of Gravitational Waves (GWs) by the LIGO array, we consider necessary to strengthen the formation of our students in Numerical Relativity (NR) in México. A key issue in the future GW astronomy is the efficiency at solving inverse problems, this means, reconstructing the physical parameters of the GW source. In the case of the Binary Black Holes (BBH) inspirals, we find the appropriate example where NR has shown its usefulness, because the waveforms predicted by NR simulations in BBH collisions are used to filter the data in the interferometer runs. The size of a catalog of such numerically generated waveforms is therefore of great importance because the more waveforms it has the easier is to reconstruct the intrinsic parameters of the BBH system. In this way, the purpose of this educative paper is to show the comparison, in terms of performance and accuracy, of the two different numerical methods used to build the catalogs, applied to the evolution of a training problem, namely the evolution of a single black hole. We present a comparison between a pseudospectral and a finite differences method in the solution of numerical general relativity equations. The system we choose to test the performance and accuracy is a spherically symmetric black hole in Eddington-Finkelstein coordinates. For the evolution we use the Einstein-Christoffel formulation of General Relativity. We compare accuracy in the violation of the Hamiltonian constraint and CPU time, both in terms of the spectral and finite differences resolution.