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

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 \emph{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 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

The phase-locked mean impulse response of a turbulent channel flow

We describe the first DNS-based measurement of the complete mean response of a turbulent channel flow to small external disturbances. Space-time impulsive perturbations are applied at one channel wall, and the linear response describes their mean effect on the flow field as a function of spatial and temporal separations. The turbulent response is shown to differ from the response a laminar flow with the turbulent mean velocity profile as base flow.Comment: Accepted for publication in Physics of Fluid

Quantum vortex reconnections

We study reconnections of quantum vortices by numerically solving the governing Gross-Pitaevskii equation. We find that the minimum distance between vortices scales differently with time before and after the vortex reconnection. We also compute vortex reconnections using the Biot-Savart law for vortex filaments of infinitesimal thickness, and find that, in this model, reconnection are time-symmetric. We argue that the likely cause of the difference between the Gross-Pitaevskii model and the Biot-Savart model is the intense rarefaction wave which is radiated away from a Gross-Pitaeveskii reconnection. Finally we compare our results to experimental observations in superfluid helium, and discuss the different length scales probed by the two models and by experiments.Comment: 23 Pages, 12 Figure

Vortex reconnections in atomic condensates at finite temperature

The study of vortex reconnections is an essential ingredient of understanding superfluid turbulence, a phenomenon recently also reported in trapped atomic Bose-Einstein condensates. In this work we show that, despite the established dependence of vortex motion on temperature in such systems, vortex reconnections are actually temperature independent on the typical length/time scales of atomic condensates. Our work is based on a dissipative Gross-Pitaevskii equation for the condensate, coupled to a semiclassical Boltzmann equation for the thermal cloud (the Zaremba-Nikuni-Griffin formalism). Comparison to vortex reconnections in homogeneous condensates further show reconnections to be insensitive to the inhomogeneity in the background density.Comment: 6 pages, 4 figure

The Inverse Power Method for the p(x)-Laplacian Problem

We present an inverse power method for the computation of the first homogeneous eigenpair of the p(x)-Laplacian problem. The operators are discretized by the finite element method. The inner minimization problems are solved by a globally convergent inexact Newton method. Numerical comparisons are made, in one- and two-dimensional domains, with other results present in literature for the constant case p(x)=p and with other minimization techniques (namely, the nonlinear conjugate gradient) for the p(x) variable case

Reliability of the time splitting Fourier method for singular solutions in quantum fluids

We study the numerical accuracy of the well-known time splitting Fourier spectral method for the approximation of singular solutions of the Gross\u2013Pitaevskii equation. In particular, we explore its capability of preserving a steady-state vortex solution, whose density profile is approximated by an accurate diagonal Pad\ue9 expansion of degree [8,8], here explicitly derived for the first time. We show by several numerical experiments that the Fourier spectral method is only slightly more accurate than a time splitting finite difference scheme, while being reliable and efficient. Moreover, we notice that, at a post-processing stage, it allows an accurate evaluation of the solution outside grid points, thus becoming particularly appealing for applications where high resolution is needed, such as in the study of quantum vortex interactions

Quasi-Newton minimization for the p(x)-Laplacian problem

We propose a quasi-Newton minimization approach for the solution of the p(x)-Laplacian elliptic problem.This method outperforms those existing for the p(x)-variable case, which are based on general purpose minimizers such as BFGS. Moreover, when compared to ad hoc techniques available in literature for the p-constant case, and usually referred to as "mesh independent", the present method turns out to be generally superior thanks to better descent directions given by the quadratic model

Spectral methods for dissipative nonlinear Schr\uf6dinger equations

In this technical report we describe an application of spectral methods to numerically solve some nonlinear Schr\uf6dinger equations with dissipative terms and carefully study the problem of vortex formation