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

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, parametrized by a `tilt' vector field . This can be used to centre the numerical stencil on the physical lightcone, by setting , where 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-Massó 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.