High-order methods for decaying two-dimensional homogeneous isotropic turbulence

TL;DR

This study compares high-order numerical schemes for simulating two-dimensional decaying turbulence, finding that certain difference methods can outperform spectral methods in accuracy and efficiency for fully resolved flows.

Contribution

It provides a comprehensive comparison of high-order difference schemes and spectral methods, highlighting their performance in long-term turbulence simulations.

Findings

Sixth-order schemes match pseudospectral accuracy.

Finite difference schemes are more computationally efficient.

Under-resolved simulations require careful scheme selection.

Abstract

Numerical schemes used for the integration of complex flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, the performance of various high-order accurate numerical schemes is investigated for direct numerical simulations (DNS) of homogeneous isotropic two-dimensional decaying turbulent flows. The numerical accuracy of compact difference, explicit central difference, Arakawa, and dispersion-relation-preserving schemes are analyzed and compared with the Fourier-Galerkin pseudospectral scheme. In addition, several explicit Runge-Kutta schemes for time integration are investigated. We demonstrate that the centered schemes suffer from spurious Nyquist signals that are generated almost instantaneously and propagate into much of the field when the numerical resolution is insufficient. We further show that the order of the scheme becomes increasingly important for increasing cell Reynolds number. Surprisingly, the sixth-order schemes are found to be in perfect agreement with the pseudospectral method. Considerable reduction in computational time compared to the pseudospectral method is also reported in favor of the finite difference schemes. Among the fourth-order schemes, the compact scheme provides better accuracy than the others for fully resolved computations. The fourth-order Arakawa scheme provides more accurate results for under-resolved computations, however, due to its conservation properties. Our results show that, contrary to conventional wisdom, difference methods demonstrate superior performance in terms of accuracy and efficiency for fully resolved DNS computations of the complex flows considered here. For under-resolved simulations, however, the choice of difference method should be made with care.