Marco Caliari

Marco Caliari, Peter Kandolf, Alexander Ostermann et al.

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.

Marco Caliari, Alexander Ostermann, Chiara Piazzola

The Schrödinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into three parts. After presenting a splitting approach for three operators with two of them being unbounded, we exemplarily prove first-order convergence of Lie splitting in this framework. The result is then applied to the magnetic Schrödinger equation, which is split into its potential, kinetic and advective parts. The latter requires special treatment in order not to lose the conservation properties of the scheme. We discuss several options. Numerical examples in one, two and three space dimensions show that the method of characteristics coupled with a nonequispaced fast Fourier transform (NFFT) provides a fast and reliable technique for achieving mass conservation at the discrete level.

Marco Caliari, Simone Zuccher

We extensively study the numerical accuracy of the well-known time splitting Fourier spectral method for the approximation of singular solutions of the Gross-Pitaevskii equation. In particular, we explore its capability of preserving a steady-state vortex solution, whose density profile is approximated by a very accurate diagonal Padé expansion of order 8, here explicitly derived for the first time. Although the Fourier spectral method turns out to be only slightly more accurate than a time splitting finite difference scheme, the former is reliable and efficient. Moreover, at a post-processing stage, it allows an accurate evaluation of the solution outside grid points, thus becoming particularly appealing when high resolution is needed, such as in the study of quantum vortex interactions.

Marco Caliari, Simone Zuccher

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.

Marco Caliari, Fabio Cassini

In this paper, we study a tensor-based method for the numerical solution of a class of diffusion--reaction equations defined on spatial domains that admit common curvilinear coordinate representations. Typical examples in 2D include disks (polar coordinates), and in 3D balls or cylinders (spherical or cylindrical coordinates) as well as spheres for problems involving the Laplace--Beltrami operator. The proposed approach is based on a carefully chosen finite difference discretization of the Laplace operators that yields matrices with a structured representation as sums of Kronecker products. For the time integration, we introduce a novel split variant of the exponential Euler method that effectively handles the stiffness and avoids the severe time step size restriction of classical explicit methods. By exploiting the peculiar form of the obtained discretized operators and the chosen splitting strategy, we compute the needed action of the $φ_1$ matrix function through suitable tensor-matrix products in a $μ$-mode framework. We demonstrate the efficiency the approach on a wide range of physically relevant 2D and 3D examples of coupled diffusion--reaction systems generating Turing patterns with up to $10^6$ degrees of freedom.

Marco Caliari, Marco Squassina

We provide some numerical computations for the soliton dynamics of the nonlinear Schrödinger equation with an external potential. After computing the ground state solution $r$ of a related elliptic equation we show that, in the semi-classical regime, the center of mass of the solution with initial datum modelled on $r$ is driven by the solution of a Newtonian type law. Finally, we provide some examples and analyze the numerical errors in the two dimensional case when $V$ is an harmonic potential.