Well-posed formulations and stable finite differencing schemes for numerical relativity

This work concerns the evolution of equations of general relativity; their mathematical properties at the continuum level, and the properties of the infinite difference scheme used to approximate their solution in numerical simulations. Stability results for finite difference approximations of partial differential equations which are first order in time and first order in space are well-known. However, systems which are first order in time and second order in space have been more successful in the field of numerical relativity than fully first order systems. For example, binary black hole simulations are accurate for much longer times. Hence, a greater understanding of the stability properties of these system is desirable. An example of such a system is the NOR (Nagy, Ortiz and Reula)[47] formulation of general relativity. We present a proof of the stability of a finite difference approximation of the linearized NOR evolution system. The new tools used to prove stability for second order in space systems are described, along with the simple example of the wave equation. In order to implement and compare different formulations of the Einstein equations in numerical simulations, the equations must be expanded from abstract tensor relations into components, discretized, and entered into a computer. This process is aided enoromously by the use of automated code generation. I present the Kranc software package which we have written to perform these tasks. It is expected, by analogy with the wave equation, that numerical simulations of systems which are of the first order in time and second order in space will be more accurate than those of fully first order systems. We present a quantitative comparison of the accuracy of formulations of the fully nonlinear Einstein equations and determine that for linearized gravitational waves, this prediction is verified. However, the same cannot be said for other test cases, and it is concluded that certain problems with the second order in space formulations make them behave worse than fully first order formulations in these cases.