  We introduce a fluid dynamics algorithm that performs with nearly spectral
accuracy, but uses finite-differences instead of FFTs to compute gradients and
thus executes 10 times faster. The finite differencing is not based on a
high-order polynomial fit. The polynomial scheme has supurb accuracy for
low-wavenumber gradients but fails at high wavenumbers. We instead use a scheme
tuned to enhance high-wavenumber accuracy at the expense of low wavenumbers,
although the loss of low-wavenumber accuracy is negligibly slight. A tuned
gradient is capable of capturing all wavenumbers up to 80 percent of the
Nyquist limit with an error of no worse than 1 percent. The fact that gradients
are based on finite differences enables diverse geometries to be considered and
eliminates the parallel communications bottleneck.
