Spectral methods for numerical relativity: The initial data problem

The partial differential equations of numerical relativity have traditionally been solved using a finite difference (FD) approximation. The accuracy of a FD solution increases as a fixed power of resolution while the computational resources required for the solution increase as the resolution raised to the (space + time) dimensionality of the problem. Modest accuracy solutions to problems involving either the initial conditions or the evolution of a dynamical black hole spacetime tax the capabilities of the computers presently available for the task, while the resources required for modest accuracy binary black hole problems are beyond what is presently available. For problems with smooth solutions alternatives to the FD approximation exist that may make more efficient use of the available computational resources. Here we investigate one of these techniques: the pseudo-spectral collocation (PSC) approximation. To determine its effectiveness relative to FD methods in solving problems in numerical relativity we use PSC to solve several two-dimensional problems that have been previously studied by other researchers using FD methods, focusing particularly on the computational resources required as a function of the desired solution accuracy. We find that PSC methods applied to these problems can achieve close to the theoretical limit of exponential convergence with problem resolution, while the computational resources required continue to scale only as the resolution raised to the problem dimensionality. Correspondingly, for solutions of even modest accuracy we find that PSC is substantially more efficient, as measured by either execution time or memory required, than FD; furthermore, these savings increase rapidly with increasing accuracy. We also discuss less quantitative but no less tangible advantages that the PSC approximation holds over the FD approximation. In particular, the solution provided by PSC is an analytic function given everywhere on the computational domain, not just at fixed grid points. Consequently, no ad hoc interpolation operators are required to determine field values at intermediate points or to evaluate the approximate solution or its derivatives on the boundaries. Since the practice of numerical relativity by finite differencing has been, and continues to be, hampered by both high computational resource demands and the difficulty of formulating acceptable finite difference alternatives to the analytic boundary conditions, we argue that PSC should be further pursued as an alternative way of formulating the computational problem of finding numerical solutions to the field equations of general relativity.