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

Accurate evaluation of divided differences for polynomial interpolation of exponential propagators

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator phi(hA) v, phi(z) = (e(z)-1)/z. In this way, it is possible to approximate.( hA) v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to "higher" order phi(k) functions, k >= 0

Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems

We consider a system of Gross-Pitaevskii equations in R^2 modelling a mixture of two Bose-Einstein condensates with repulsive interaction. We aim to study the qualitative behaviour of ground and excited state solutions. We allow two different harmonic and off-centered trapping potentials and study the spatial patterns of the solutions within the Thomas-Fermi approximation as well as phase segregation phenomena within the large-interaction regime.Comment: 21 pages, to appear in Dyn. Partial Diff. Equa

On a bifurcation value related to quasi-linear Schrodinger equations

By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the quasi-linear Schrodinger equation.Comment: 9 page

The Leja method revisited: backward error analysis for the matrix exponential

The Leja method is a polynomial interpolation procedure that can be used to compute matrix functions. In particular, computing the action of the matrix exponential on a given vector is a typical application. This quantity is required, e.g., in exponential integrators. The Leja method essentially depends on three parameters: the scaling parameter, the location of the interpolation points, and the degree of interpolation. We present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy. Additional aspects that are required for an efficient and reliable implementation are discussed. Numerical examples that illustrate the performance of our Matlab code are included

Direction splitting of φ\varphi-functions in exponential integrators for dd-dimensional problems in Kronecker form

In this manuscript, we propose an efficient, practical and easy-to-implement way to approximate actions of $\varphi$-functions for matrices with $d$-dimensional Kronecker sum structure in the context of exponential integrators up to second order. The method is based on a direction splitting of the involved matrix functions, which lets us exploit the highly efficient level 3 BLAS for the actual computation of the required actions in a $\mu$-mode fashion. The approach has been successfully tested on two- and three-dimensional problems with various exponential integrators, resulting in a consistent speedup with respect to a technique designed to compute actions of $\varphi$-functions for Kronecker sums

A second order directional split exponential integrator for systems of advection–diffusion–reaction equations

We propose a second order exponential scheme suitable for two-component coupled systems of stiff evolutionary advection–diffusion–reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension. Several numerical examples in 2D and 3D with physically relevant (advective) Schnakenberg, FitzHugh–Nagumo, DIB, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques

A second order directional split exponential integrator for systems of advection--diffusion--reaction equations

We propose a second order exponential scheme suitable for two-component coupled systems of stiff advection--diffusion--reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small-sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension $d$. Several numerical experiments in 2D and 3D with physically relevant DIB, Schnakenberg, FitzHugh--Nagumo, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques