  We present an efficient high-precision numerical approach for the
Davey-Stewartson (DS) II equation, treating initial data from the Schwartz
class of smooth, rapidly decreasing functions. As with previous approaches, the
presented code uses discrete Fourier transforms for the spatial dependence and
Driscoll's composite Runge-Kutta method for the time dependence. Since the DS
equation is a nonlocal, nonlinear Schr\"odinger equation with a singular symbol
for the nonlocality, standard Fourier methods in practice only reach accuracies
of the order of $10^{-6}$ or less for typical examples. This was previously
demonstrated for the defocusing case by comparison with a numerical approach
for DS via inverse scattering. By applying a regularization to the singular
symbol, originally developed for D-bar problems, the presented code is shown to
reach machine precision. The code can treat integrable and non-integrable DS II
equations. Moreover, it has the same %complexity numerical complexity as
existing codes for DS II. Several examples for the integrable defocusing DS II
equation are discussed as test cases.
  In an appendix by C.~Kalla, a doubly periodic solution to the defocusing DS
II equation is presented, providing a test for direct DS codes based on Fourier
methods.
