Numerical stability of the Z4c formulation of general relativity

TL;DR

This paper investigates the numerical stability of the Z4c formulation of general relativity, demonstrating a new stable discretization method and comparing its performance with BSSNOK in various tests, showing improved constraint control and stability.

Contribution

The paper introduces a novel discretization for tensors in Z4c and provides comprehensive stability analysis and comparisons with BSSNOK in non-linear regimes.

Findings

The new discretization is formally numerically stable in linear regimes.

Z4c evolutions show lower constraint violations than BSSNOK in certain tests.

Z4c maintains convergence over more light-crossing times than BSSNOK.

Abstract

We study numerical stability of different approaches to the discretization of a conformal decomposition of the Z4 formulation of general relativity. We demonstrate that in the linear, constant coefficient regime a novel discretization for tensors is formally numerically stable with a method of lines time-integrator. We then perform a full set of apples with apples tests on the non-linear system, and thus present numerical evidence that both the new and standard discretizations are, in some sense, numerically stable in the non-linear regime. The results of the Z4c numerical tests are compared with those of BSSNOK evolutions. We typically do not employ the Z4c constraint damping scheme and find that in the robust stability and gauge wave tests the Z4c evolutions result in lower constraint violation at the same resolution as the BSSNOK evolutions. In the gauge wave tests we find that the Z4c evolutions maintain the desired convergence factor over many more light-crossing times than the BSSNOK tests. The difference in the remaining tests is marginal.