Causal differencing of flux-conservative equations applied to black hole spacetimes

TL;DR

This paper introduces a causal differencing scheme for flux-conservative PDEs, enabling stable, long-term simulations of black hole spacetimes by aligning numerical stencils with the physical light cone.

Contribution

It develops a flexible finite-difference method that can be oriented arbitrarily, including on the light cone, and applies it to Einstein equations for black hole evolution.

Findings

Stable long-term evolution of black holes achieved

Causal differencing aligns numerical and physical light cones

Method applicable to hyperbolic flux-conservative equations

Abstract

We give a general scheme for finite-differencing partial differential equations in flux-conservative form to second order, with a stencil that can be arbitrarily tilted with respect to the numerical grid, parameterized by a "tilt" vector field gamma^A. This can be used to center the numerical stencil on the physical light cone, by setting gamma^A = beta^A, where beta^A is the usual shift vector in the 3+1 split of spacetime, but other choices of the tilt may also be useful. We apply this "causal differencing" algorithm to the Bona-Masso equations, a hyperbolic and flux-conservative form of the Einstein equations, and demonstrate long term stable causally correct evolutions of single black hole systems in spherical symmetry.