Aliasing and phase shifting in pseudo-spectral simulations of the incompressible Navier-Stokes equations

TL;DR

This paper analyzes phase-shifting dealiasing in pseudo-spectral Navier-Stokes simulations, demonstrating its efficiency and providing the first open-source implementation, with significant speedups over traditional methods.

Contribution

It offers a comprehensive analysis, derivation, and comparison of phase-shifting algorithms, and introduces an open-source implementation in Fluidsim.

Findings

Phase-shifting methods achieve up to 3x speedup over RK4 with 2/3 truncation.

Phase-shifting cancels aliasing errors effectively depending on the time-stepping scheme.

The implementation is publicly available in the Fluidsim framework.

Abstract

Pseudo-spectral methods are widely used for direct numerical simulations of turbulence, but the standard 2/3 truncation rule for dealiasing is computationally expensive -- accounting for up to 80% of the total cost in three dimensions. Phase shifting methods provide a more efficient alternative by canceling aliasing errors the combination of nonlinear terms evaluated on shifted grids, allowing the same physical resolution to be achieved on a coarser numerical grid. Despite their use in high-resolution turbulence codes, these methods remain poorly documented in the literature and no open-source implementation exists. This paper presents a comprehensive analysis of phase-shifting dealiasing for pseudo-spectral simulations of the incompressible Navier-Stokes equations. We derive the aliasing mechanism from quadratic nonlinearities in discrete Fourier space and explain how phase-shifting cancels aliasing contributions exactly or approximately depending on the time-stepping scheme. We describe and compare several algorithms -- including the exact and approximate RK2 phase-shifting schemes of Patterson Jr and Orszag (1971) and Rogallo (1981), and an extension to forced flows -- and discuss their interaction with different truncation geometries in three dimensions. All algorithms are implemented in the open-source framework Fluidsim, providing the first publicly available implementation of phase-shifting dealiasing for pseudo spectral Navier-Stokes solvers. We evaluate the methods on two test cases: the transition to turbulence of Taylor-Green vortices and forced homogeneous isotropic turbulence at $Re_\lambda = 200$. Phase-shifting methods achieve speedups of up to a factor of 3 compared to RK4 with 2/3 truncation at the same maximum wavenumber, with small accuracy loss.