Constraint-preserving boundary conditions in the Z4 Numerical Relativity formalism

TL;DR

This paper develops and analyzes boundary conditions that preserve constraints in the Z4 formalism of Numerical Relativity, ensuring stable and consistent simulations through Sommerfeld and dissipative boundary conditions.

Contribution

It introduces constraint-preserving boundary conditions for the Z4 system and establishes stability criteria, including conditions for Sommerfeld and maximally dissipative types.

Findings

Constraint-preserving boundary conditions are derived for Z4.

A necessary and sufficient stability condition is identified for Sommerfeld conditions.

Maximally dissipative boundary conditions are shown to be stable even with corners and edges.

Abstract

The constraint-preserving approach, which aim is to provide consistent boundary conditions for Numerical Relativity simulations, is discussed in parallel with other recent developments. The case of the Z4 system is considered, and constraint-preserving boundary conditions of the Sommerfeld type are provided. A necessary condition for the stability of the proposed boundary conditions is obtained, which amounts to the requirement of a symmetric ordering of space derivatives. This requirement is numerically seen to be also sufficient in the absence of corners and edges. Maximally dissipative boundary conditions are also implemented. In this case, a less restrictive stability condition is obtained, which is shown numerically to be also sufficient even in the presence of corners and edges.