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

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 $Re_λ= 200$. 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.

Figures (23)

• Figure 1: Illustration of aliasing generated by nonlinear terms in one-dimensional spectral space, adapted from Rogallo1981, with $2k_N = (2\pi / L) N$ and $k_m + k_n = k_{\mathrm{nl}}$.
• Figure 2: Common truncations used without phase shifting. Modes are set to zero in the gray regions. In Fluidsim, (a) is referred to as "cubic" and (b) as "spherical".
• Figure 3: Numerical and exact solutions of Eq. \ref{['eq:model-1d']} in spectral space after one RK2 scheme time step. The arrows indicate how unresolved modes are aliased (see Fig. \ref{['fig:illu_alias_1d']}).
• Figure 4: Temporal evolution of (a) total kinetic energy $E(t)$ (solid line) and of the parts associated with $v_x$ (dashed lines) and $v_z$ (dotted lines) (b) energy dissipation, (c) energy relative error, and (d) energy dissipation relative error for dealiased ($C_t=2/3$) and aliased ($C_t=1$) simulations at $k_{\max} \tilde{\eta} \simeq 0.7$ (see Table \ref{['tab:192']}).
• Figure 5: Spectral statistics averaged between $t=9$ and $t=14$ for dealiased and aliased simulations for $k_{\max} \tilde{\eta} \simeq 0.7$ (see Table \ref{['tab:192']}). (a) Compensated 3D energy spectra (versus wavenumber modulus $k$, dotted lines) and 1D energy spectra (versus $k_x$, solid lines). (b) Nonlinear energy flux (solid lines) and cumulated dissipation energy (dashed lines).