Constrained Transport Algorithms for Numerical Relativity. I. Development of a Finite‐Difference Scheme

A scheme is presented for accurately propagating the gravitational fieldconstraints in finite difference implementations of numerical relativity. Themethod is based on similar techniques used in astrophysicalmagnetohydrodynamics and engineering electromagnetics, and has properties of afinite differential calculus on a four-dimensional manifold. It is motivated bythe arguments that 1) an evolutionary scheme that naturally satisfies theBianchi identities will propagate the constraints, and 2) methods in whichtemporal and spatial derivatives commute will satisfy the Bianchi identitiesimplicitly. The proposed algorithm exactly propagates the constraints in alocal Riemann normal coordinate system; {\it i.e.}, all terms in the Bianchiidentities (which all vary as $\partial^3 g$) cancel to machine roundoffaccuracy at each time step. In a general coordinate basis, these terms, andthose that vary as $\partial g\partial^2 g$, also can be made to cancel, butdifferences of connection terms, proportional to $(\partial g)^3$, will remain,resulting in a net truncation error. Detailed and complex numerical experimentswith four-dimensional staggered grids will be needed to completely examine thestability and convergence properties of this method. If such techniques are successful for finite difference implementations ofnumerical relativity, other implementations, such as finite element (andeventually pseudo-spectral) techniques, might benefit from schemes that usefour-dimensional grids and that have temporal and spatial derivatives thatcommute.Comment: 27 pages, 5 figure