Constraint-preserving boundary conditions in the 3+1 first-order approach

TL;DR

This paper introduces energy-momentum constraint-preserving boundary conditions for the first-order Z4 formalism, demonstrating stability and convergence in numerical tests, and extends symmetric hyperbolicity conditions.

Contribution

It proposes new boundary conditions that preserve constraints in the first-order Z4 approach and provides a new symmetrizer extending hyperbolicity domain.

Findings

Boundary conditions maintain low constraint violations.

Numerical implementation shows stable and convergent results.

New symmetrizer extends hyperbolicity domain.

Abstract

A set of energy-momentum constraint-preserving boundary conditions is proposed for the first-order Z4 case. The stability of a simple numerical implementation is tested in the linear regime (robust stability test), both with the standard corner and vertex treatment and with a modified finite-differences stencil for boundary points which avoids corners and vertices even in cartesian-like grids. Moreover, the proposed boundary conditions are tested in a strong field scenario, the Gowdy waves metric, showing the expected rate of convergence. The accumulated amount of energy-momentum constraint violations is similar or even smaller than the one generated by either periodic or reflection conditions, which are exact in the Gowdy waves case. As a side theoretical result, a new symmetrizer is explicitly given, which extends the parametric domain of symmetric hyperbolicity for the Z4 formalism. The application of these results to first-order BSSN-like formalisms is also considered.