  Time integration of Fourier pseudo-spectral DNS is usually performed using
the classical fourth-order accurate Runge--Kutta method, or other methods of
second or third order, with a fixed step size. We investigate the use of
higher-order Runge-Kutta pairs and automatic step size control based on local
error estimation. We find that the fifth-order accurate Runge--Kutta pair of
Bogacki \& Shampine gives much greater accuracy at a significantly reduced
computational cost. Specifically, we demonstrate speedups of 2x-10x for the
same accuracy. Numerical tests (including the Taylor-Green vortex,
Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the
reliability and efficiency of the method. We also show that adaptive time
stepping provides a significant computational advantage for some problems (like
the development of a Rayleigh-Taylor instability) without compromising
accuracy.
