Cauchy-perturbative matching revisited: tests in spherical symmetry

TL;DR

This paper demonstrates that advanced numerical techniques enable highly accurate and long-term Cauchy-perturbative matching in spherical symmetry, significantly reducing errors in black hole simulations over extended periods.

Contribution

It introduces the use of high-order summation-by-parts operators, penalty boundaries, and constraint-preserving boundary conditions to improve CPM accuracy in spherical symmetry.

Findings

Errors in CPM are very small with new techniques.

Long-term stable evolutions of black holes are achievable.

Error of 0.3% after 1,000,000 M evolution time.

Abstract

During the last few years progress has been made on several fronts making it possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity in a more robust and accurate way. This paper is the first in a series where we plan to analyze CPM in the light of these new results. Here we start by testing high-order summation-by-parts operators, penalty boundaries and contraint-preserving boundary conditions applied to CPM in a setting that is simple enough to study all the ingredients in great detail: Einstein's equations in spherical symmetry, describing a black hole coupled to a massless scalar field. We show that with the techniques described above, the errors introduced by Cauchy-perturbative matching are very small, and that very long term and accurate CPM evolutions can be achieved. Our tests include the accretion and ring-down phase of a Schwarzschild black hole with CPM, where we find that the discrete evolution introduces, with a low spatial resolution of \Delta r = M/10, an error of 0.3% after an evolution time of 1,000,000 M. For a black hole of solar mass, this corresponds to approximately 5 s, and is therefore at the lower end of timescales discussed e.g. in the collapsar model of gamma-ray burst engines. (abridged)