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

TL;DR

This paper demonstrates that using higher-order Runge-Kutta methods with adaptive step sizing significantly improves the efficiency and accuracy of Fourier pseudo-spectral DNS for incompressible turbulence simulations.

Contribution

It introduces the use of fifth-order Runge-Kutta pairs with adaptive step size control to enhance DNS time integration efficiency and accuracy.

Findings

Speedups of 2x-10x for the same accuracy

Reliable and efficient method confirmed by numerical tests

Adaptive time stepping benefits complex turbulence simulations

Abstract

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.