Relative accuracy of turbulence simulations using pseudo-spectral and finite difference solvers

TL;DR

This study compares spectral and finite-difference turbulence simulations, finding that despite spectral methods being more accurate per timestep, both yield similar results in turbulence modeling due to error cancellation within the attractor.

Contribution

The paper demonstrates that finite-difference and spectral solvers produce nearly identical turbulence simulation results, challenging the assumption that spectral methods are always more accurate in practice.

Findings

Both methods produce similar energy spectra and flow profiles.

Numerical errors tend to cancel within the turbulence attractor.

Finite-difference methods are more versatile and efficient for large grids.

Abstract

For a single timestep, a spectral solver is known to be more accurate than its finite-difference counterpart. However, as we show in this paper, turbulence simulations using the two methods have nearly the same accuracy. In this paper, we simulate forced hydrodynamic turbulence on a uniform 256$^3$ grid for Reynolds numbers 965, 1231, 1515, and 1994. We show that the two methods yield nearly the same evolution for the total energy and the flow profiles. In addition, the steady-state energy spectrum, energy flux, and probability distribution functions of the velocity and its derivatives are very similar. We argue that within a turbulence attractor, the numerical errors are likely to get cancelled (rather than get added up), which leads to similar results for the finite-difference and spectral methods. These findings are very valuable, considering that a parallel finite-difference simulation is more versatile and efficient (for large grids) than its spectral counterpart.