More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence

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.