Well-balanced high order finite difference WENO schemes for a first-order Z4 formulation of the Einstein field equations

TL;DR

This paper develops high-order well-balanced finite difference WENO schemes tailored for the first-order Z4 formulation of Einstein's equations, enhancing stability and accuracy in numerical relativity simulations.

Contribution

It introduces a novel class of high-order well-balanced WENO methods applicable to non-conservative first-order Einstein equations, improving long-term stability and constraint preservation.

Findings

Achieved stable long-term evolution of stationary solutions.

Enhanced accuracy in numerical solutions of Einstein equations.

Demonstrated effectiveness of the schemes in numerical relativity simulations.

Abstract

In this work we aim at developing a new class of high order accurate well-balanced finite difference (FD) Weighted Essentially Non-Oscillatory (WENO) methods for numerical general relativity, which can be applied to any first-order reduction of the Einstein field equations, even if non-conservative terms are present. We choose the first-order non-conservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. Upon the introduction of auxiliary variables, the vacuum Einstein equations in first order form constitute a ...