Spectral Methods for Numerical Relativity. The Initial Data Problem

TL;DR

This paper explores pseudospectral collocation (PSC) as a highly efficient alternative to finite differencing for solving elliptic equations in numerical relativity, demonstrating exponential convergence and computational advantages.

Contribution

It introduces PSC for solving the Hamiltonian constraint in black hole spacetimes, showing its efficiency and accuracy advantages over traditional finite differencing methods.

Findings

PSC converges exponentially to the exact solution with increasing basis functions

PSC is more efficient in time and memory than finite differencing

No boundary interpolation or special boundary conditions are needed with PSC

Abstract

Numerical relativity has traditionally been pursued via finite differencing. Here we explore pseudospectral collocation (PSC) as an alternative to finite differencing, focusing particularly on the solution of the Hamiltonian constraint (an elliptic partial differential equation) for a black hole spacetime with angular momentum and for a black hole spacetime superposed with gravitational radiation. In PSC, an approximate solution, generally expressed as a sum over a set of orthogonal basis functions (e.g., Chebyshev polynomials), is substituted into the exact system of equations and the residual minimized. For systems with analytic solutions the approximate solutions converge upon the exact solution exponentially as the number of basis functions is increased. Consequently, PSC has a high computational efficiency: for solutions of even modest accuracy we find that PSC is substantially more efficient, as measured by either execution time or memory required, than finite differencing; furthermore, these savings increase rapidly with increasing accuracy. The solution provided by PSC is an analytic function given everywhere; consequently, no interpolation operators need to be defined to determine the function values at intermediate points and no special arrangements need to be made to evaluate the 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, 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.