Constrained Transport Algorithms for Numerical Relativity. I. Development of a FiniteDifference Scheme
Abstract
A scheme is presented for accurately propagating the gravitational field constraints in finitedifference implementations of numerical relativity. The method is based on similar techniques used in astrophysical magnetohydrodynamics and engineering electromagnetics and has properties of a finite differential calculus on a fourdimensional manifold. It is motivated by the arguments that (1) an evolutionary scheme that naturally satisfies the Bianchi identities will propagate the constraints and (2) methods in which temporal and spatial derivatives commute will satisfy the Bianchi identities implicitly. The proposed algorithm exactly propagates the constraints in a local Riemann normal coordinate system; i.e., all terms in the Bianchi identities (which all vary as ∂^{3}g) cancel to machine roundoff accuracy at each time step. In a general coordinate basis, these terms and those that vary as ∂g∂^{2}g also can be made to cancel, but differences of connection terms, proportional to (∂g)^{3}, will remain, resulting in a net truncation error. Detailed and complex numerical experiments with fourdimensional staggered grids will be needed to completely examine the stability and convergence properties of this method. If such techniques are successful for finitedifference implementations of numerical relativity, other implementations, such as finiteelement (and eventually pseudospectral) techniques, might benefit from schemes that use fourdimensional grids and that have temporal and spatial derivatives that commute.
 Publication:

The Astrophysical Journal
 Pub Date:
 October 2003
 DOI:
 10.1086/377166
 arXiv:
 arXiv:astroph/0312052
 Bibcode:
 2003ApJ...595..980M
 Keywords:

 Black Hole Physics;
 Relativity;
 Astrophysics;
 General Relativity and Quantum Cosmology;
 Physics  Computational Physics
 EPrint:
 27 pages, 5 figures