Fourier continuation method for incompressible fluids with boundaries

Abstract

We present a Fourier Continuation-based parallel pseudospectral method for incompressible fluids in cuboid non-periodic domains. The method produces dispersionless and dissipationless derivatives with fast spectral convergence inside the domain, and with very high order convergence at the boundaries. Incompressibility is imposed by solving a Poisson equation for the pressure. Being Fourier-based, the method allows for fast computation of spectral transforms. It is compatible with uniform grids (although refined or nested meshes can also be implemented), which in turn allows for explicit time integration at sufficiently high Reynolds numbers. Using a new parallel code named SPECTER we illustrate the method with two problems: channel flow, and plane Rayleigh-Bénard convection under the Boussinesq approximation. In both cases the method yields results compatible with previous studies using other high-order numerical methods, with mild requirements on the time step for stability.

Vector potential-based MHD solver for non-periodic flows using Fourier continuation expansions

“…That is not the case, however, for the non-periodic z dimension, as the well known Gibbs ringing phenomenon [37] would severely degrade accuracy in the representation of the fields' derivatives. To circumvent this limitation we employ a computationally efficient continuation methodology, known as FC-Gram, first introduced in [38,16] and recently utilized in [19] for a hydrodynamic Navier-Stokes solver.…”

“…Besides yielding high order accuracy, this technique produces spatially dispersionless derivatives [18] -in view of its reliance on Fourier expansions -and hence phase speed propagation errors might arise solely as a result of the time stepping strategy. Another advantage of the FC-Gram method is that Poisson equations, which are common in some formulations of the incompressible hydrodynamic equations, can be easily and efficiently solved in bounded domains [19]. As a result, to the present day the FC-Gram method has been successfully employed in a wide range of PDE problems [17,20,21,19].…”

“…Another advantage of the FC-Gram method is that Poisson equations, which are common in some formulations of the incompressible hydrodynamic equations, can be easily and efficiently solved in bounded domains [19]. As a result, to the present day the FC-Gram method has been successfully employed in a wide range of PDE problems [17,20,21,19].…”

Correlation between Buoyancy Flux, Dissipation and Potential Vorticity in Rotating Stratified Turbulence

“…It parallelizes efficiently using a combination of MPI, Open-MP as well as CUDA [22]. It also has a version with non-periodic boundary conditions in the vertical [23]. We take initial conditions that are typical for homogeneous isotropic turbulence (HIT), with zero temperature fluctuations (thus, zero initial potential energy), and velocity modes centered on the large scales, 2 ≤ |k 0 | ≤ 3.…”

Interplay between turbulence and waves: large-scale helical transfer, and small-scale dissipation and mixing in fluid and Hall-MHD turbulence

“…In all the results given in this paper, the GHOST code (Geophysical High-Order Suite for Turbulence) was used; it is a solver with spectral accuracy for a variety of fluid and plasma equations. It parallelizes e ciently, including using [48]; it includes the possibility of non-unity aspect ratio, and it now allows for nonperiodic boundary conditions in one direction through a Fourier continuation method [49]. Furthermore, the Boussinesq equations (in the absence of magnetic field), with ✓ the temperature fluctuations around a mean profile, and normalized so as to have the physical dimensions of a velocity, are:…”