  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.
