INFFTM: Fast evaluation of 3d Fourier series in MATLAB with an application to quantum vortex reconnections

Abstract

Although Fourier series approximation is ubiquitous in computational physics owing to the Fast Fourier Transform (FFT) algorithm, efficient techniques for the fast evaluation of a three-dimensional truncated Fourier series at a set of arbitrary points are quite rare, especially in MATLAB language. Here we employ the Nonequispaced Fast Fourier Transform (NFFT, by J. Keiner, S. Kunis, and D. Potts), a C library designed for this purpose, and provide a Matlab R and GNU Octave interface that makes NFFT easily available to the Numerical Analysis community. We test the effectiveness of our package in the framework of quantum vortex reconnections, where pseudospectral Fourier methods are commonly used and local high resolution is required in the post-processing stage. We show that the efficient evaluation of a truncated Fourier series at arbitrary points provides excellent results at a computational cost much smaller than carrying out a numerical simulation of the problem on a sufficiently fine regular grid that can reproduce comparable details of the reconnecting vortices.

Twist effects in quantum vortices and phase defects

“…GPE(1) is integrated numerically by employing the well-known second-order Strang splitting Fourier spectral method (Koplik and Levine 1993), that has been consistently used for numerical simulations of vortex reconnection (Zuccher et al 2012, Allen et al 2014, Zuccher and Ricca 2015, Caliari and Zuccher 2017. Its main advantages are that mass conservation is enforced exactly and efficiency is guaranteed by fast Fourier transform algorithms.…”

Relaxation of twist helicity in the cascade process of linked quantum vortices

“…[19]), a package originally written in C language and recently included in the interface INFFTM developed for MATLAB and GNU OCTAVE (see Ref. [20] for details).…”

“…Arc-length derivatives are thus computed without any further regularization to obtain, for instance, the local Frenet triad {T,N,B} given by the unit tangent, normal, and binormal vector. The ribbon edge C * is obtained from the tube axis C (the ribbon baseline) in four steps: (i) a set of equispaced points is generated on the circle of radius = ξ/2 = 0.5, centered on C and perpendicular toT; (ii) at each point, ψ is evaluated by using INFFTM [20]; (iii) the phase is then interpolated at a constant valueθ = π ; (iv) the ribbon edge C * , placed at = constant from C on the isophase surfaceθ = π , is computed and width renormalized to 1. By construction the ribbon is thus given by a smoothly varying function of arc length, that is then used to compute T w by employing Eq.…”

Identification of vortices in quantum fluids: Finite element algorithms and programs

“…The main advantage of the toolbox distributed with this paper is its ability to give fast and accurate results for vortex identification in different configurations: 2D, 3D and experimental images. While more precise numerical methods for the identification of vortices could be set by combining local minimization and Fourier interpolation (Villois et al, 2016;Caliari and Zuccher, 2017), we showed that the the computation of the zeros of a P1 finite element function is a reliable method to obtain a good degree of precision while keeping the computational cost low. This approach is well suited for fields with a large number of vortices, as obtained in high resolution simulation of quantum turbulence, where local minimization would be too costly.…”

“…Precise methods for the tracking of vortices in complex fields were developed by Villois et al (2016) using a Newton-Raphson method to determine the zeros of the wave function combined with Fourier interpolation for the evaluation of the wave function outside grid points. Caliari and Zuccher (2017) suggested another method adapted to spectral Fourier computations, based on non-equispaced Fast Fourier Transform to truncate Fourier series at arbitrary points and finally identify vortex line positions during vortex reconnection. A fast real-time visualisation technique of vortices in QT simulations of high resolution simulations was developed by Liu et al (2020) using a graph-based method.…”