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This paper develops a method for solving Einstein's equation numerically onmulti-cube representations of manifolds with arbitrary spatial topologies. Thismethod is designed to provide a set of flexible, easy to use computationalprocedures that make it possible to explore the never before studied propertiesof solutions to Einstein's equation on manifolds with arbitrary toplogicalstructures. A new covariant, first-order symmetric-hyperbolic representation ofEinstein's equation is developed for this purpose, along with the neededboundary conditions at the interfaces between adjoining cubic regions.Numerical tests are presented that demonstrate the long-term numericalstability of this method for evolutions of a complicated, time-dependentsolution of Einstein's equation coupled to a complex scalar field on a manifoldwith spatial topology S^3. The accuracy of these numerical test solutions isevaluated by performing convergence studies and by comparing the fullnon-linear numerical results to the analytical perturbation solutions, whichare also derived here.

Solving Einstein's Equation Numerically on Manifolds With Arbitrary Spatial Topologies