Generalized harmonic spatial coordinates and hyperbolic shift conditions

TL;DR

This paper introduces a generalized harmonic spatial coordinate condition with a shift evolution equation, analyzes its mathematical properties, and demonstrates its effectiveness through numerical experiments in general relativity.

Contribution

It proposes a new generalized harmonic shift condition, making it covariant and suitable for various coordinate systems, and studies its impact on hyperbolicity and numerical stability.

Findings

The generalized shift condition can be decoupled from slicing using a rescaled shift vector.

The proposed condition maintains hyperbolicity in 1+1 and 3+1 spherical symmetry.

Numerical experiments show stable behavior of the shift condition in simulations.

Abstract

We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blow-ups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.