A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose-Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin-Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate

Numerical investigation on nonlocal problems with the fractional Laplacian

Nonlocal models have recently become a powerful tool for studying complex systems with long-range interactions or memory effects, which cannot be described properly by the traditional differential equations. So far, different nonlocal (or fractional differential) models have been proposed, among which models with the fractional Laplacian have been well applied. The fractional Laplacian (-Δ)α/2 represents the infinitesimal generator of a symmetric α-stable Lévy process. It has been used to describe anomalous diffusion, turbulent flows, stochastic dynamics, finance, and many other phenomena. However, the nonlocality of the fractional Laplacian introduces considerable challenges in its mathematical modeling, numerical simulations, and mathematical analysis. To advance the understanding of the fractional Laplacian, two novel and accurate finite difference methods -- the weighted trapezoidal method and the weighted linear interpolation method are developed for discretizing the fractional Laplacian. Numerical analysis is provided for the error estimates, and fast algorithms are developed for their efficient implementation. Compared to the current state-of-the-art methods, these two methods have higher accuracy but less computational complexity. As an application, the solution behaviors of the fractional Schördinger equation are investigated to understand the nonlocal effects of the fractional Laplacian. First, the eigenvalues and eigenfunctions of the fractional Schrödinger equation in an infinite potential well are studied, and the results provide insights into an open problem in the fractional quantum mechanics. Second, three Fourier spectral methods are developed and compared in solving the fractional nonlinear Schördinger equation (NLS), among which the SSFS method is more effective in the study of the plane wave dynamics. Sufficient conditions are provided to avoid the numerical instability of the SSFS method. In contrast to the standard NLS, the plane wave dynamics of the fractional NLS are more chaotic due to the long-range interactions --Abstract, page iii

LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I

CONTENTS J. Bona Derivation and some fundamental properties of nonlinear dispersive waves equations F. Planchon Schr\"odinger equations with variable coecients P. Rapha\"el On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio

Magnus-based geometric integrators for dynamical systems with time-dependent potentials

[ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial. La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo. El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis. El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo. La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético. En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock. El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente. El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura.[CA] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest. L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps. El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes fonamentals utilitzades al llarg de la tesi. El disseny dels integradors proposats es basa en els mètodes de composició i escissió i en el desenvolupament de Magnus. En el Capítol 3, es descriu el primer. La seua idea principal consta d'una recombinació d'uns integradors senzills per a obtenir la solució del problema. El concepte important de les condicions d'orde es descriu en eixe capítol. El Capítol 4 fa un resum de les àlgebres de Lie i del desenvolupament de Magnus que són les ferramentes algebraiques que permeten expressar la solució d'equacions diferencials dependents del temps. L'equació lineal de Schrödinger amb potencial dependent del temps està examinada en el Capítol 5. Donat la seua estructura particular, nous mètodes quasi sense commutadors, basats en el desenvolupament de Magnus, són construïts. La seua eficiència és demostrada en uns experiments numèrics amb el model de Walker-Preston d'una molècula dins d'un camp electromagnètic. En el Capítol 6 es dissenyen els mètodes de Magnus-escissió per a les equacions d'onda i de Hill. El seu rendiment està demostrat en els experiments numèrics amb diversos sistemes oscil·latoris: amb l'equació de Mathieu, l'ec. de Hill matricial, les equacions d'onda i de Klein-Gordon-Fock. El Capítol 7 explica com l'enfocament algebraic i el desenvolupament de Magnus poden generalitzar-se als problemes no lineals. L'exemple utilitzat és el problema de Kepler amb massa decreixent. El Capítol 8 conclou la tesi, ressenya els resultats i traça les possibles direccions de la investigació futura.[EN] The present thesis addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics because they come from quantum, classical and celestial mechanics. The goal of the thesis is to construct integrators for several import ant non-autonomous problems: the Schrödinger equation, which is the cornerstone of quantum mechanics; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass. Chapter 1 describes the motivation and the aims of the work in the historical context of numerical integration. In Chapter 2 essential concepts and some fundamental tools used throughout the thesis are introduced. The design of the proposed integrators is based on the composition and splitting methods and the Magnus expansion. In Chapter 3, the former is described. Their main idea is to recombine some simpler integrators to obtain the solution. The salient concept of order conditions is described in that chapter. Chapter 4 summarises Lie algebras and the Magnus expansion ¿ algebraic tools that help to express the solution of time-dependent differential equations. The linear Schrödinger equation with time-dependent potential is considered in Chapter 5. Given its particular structure, new, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker-Preston model of a molecule in an electromagnetic field. In Chapter 6, Magnus-splitting methods for the wave and the Hill equations are designed. Their performance is demonstrated in numerical experiments with various oscillatory systems: the Mathieu equation, the matrix Hill eq., the wave and the Klein-Gordon-Fock eq. Chapter 7 shows how the algebraic approach and the Magnus expansion can be generalised to non-linear problems. The example used is the Kepler problem with decreasing mass. The thesis is concluded by Chapter 8, in which the results are reviewed and possible directions of future work are outlined.Kopylov, N. (2019). Magnus-based geometric integrators for dynamical systems with time-dependent potentials [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/118798TESI

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

This paper introduces and analyses the new grid-based tensor approach to approximate solution of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree-Fock equation over a spatial $L_1\times L_2\times L_3$ lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large $L_1\times L_2\times L_3$ lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic $L\times 1\times 1$ lattice chain in a 3D rectangular "tube" with $L$ up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large $L$.Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1408.383

Numerical Integrators for Physical Applications

In this thesis, we report on our work in two very fundamental fields of physics which still have not been merged in a satisfactory way by a combining physical theory. One area is the field of very small particles, most accurately described by quantum mechanics. Here, we are interested in the phenomenon of superconductivity. The other area is that of the very heavy objects of our universe. Their most fundamental description is based on the theory of general relativity. Our particular interest lies in binary systems of compact objects rotating around each other, constantly radiating gravitational waves in the process. Although quantum mechanics and general relativity are worlds apart from a physical point of view, they inhibit some analogies when seen from our numerical perspective. For our aim is the same in both fields: We want to provide numerical tools for the simulations of interesting physical processes. Regarding binary systems we want to compare two descriptions of their motion in space. The first is given by the Mathisson--Papapetrou equations. In order to study the evolution as given by these equations, we develop an efficient integration scheme based on Gauss Runge--Kutta methods. An intriguing challenge is given by the fact that part of the equations of motion have only be given implicitly. All obstacles notwithstanding, we present an efficient integrator which preserves the constants of motion even over long times. The second description of a binary's motion is given by a Hamiltonian approximation of the Mathisson--Papapetrou equations. We want to study whether this prescription yields physically valid results. To this aim, we first come up with an efficient numerical evolution scheme, again recurring to Gauss Runge-Kutta integrators. Our scheme conserves the Hamiltonian structure, thus yielding reliable results for long time spans. Then, we test the Hamiltonian approach in different aspects. When studying the behavior of important constants of motion, we have found out that the Hamiltonian in its originally published form must be based on unphysical assumptions. This triggered new theoretical studies by our collaborators from physics with the aim of finding better suited alternatives. Their new results and suggestions are tested with the help of our algorithms. The --now physically reasonable-- Hamiltonian descriptions are well-suited to investigate the binary systems for chaos with the help of surface sections. Hence, we take use of the collocation property of the Gauss Runge--Kutta schemes to present an accurate and convenient algorithm for the calculation of such sections. In the realm of superconductivity, we consider the time-dependent BCS equations. These are quite involved partial differential equations describing the evolution of the Cooper pair density within a superconducting material or a superfluid. A very hot topic in the theoretical physics community concerns the question as to whether there exists, close to the critical temperature, a more convenient equation for a reliable approximation on a macroscopic scale. We take on this question from a numerical point of view. For this, we compare the evolution of a system with contact interaction given by the BCS equations to the one obtained via a linearized approximation by means of a thorough numerical study. We concentrate on a translation invariant system and develop two new numerical solvers based on so-called splitting methods. Splitting the coupled equations into more convenient subproblems and aptly combining the partial results, we come up with efficient and accurate schemes whose CPU times depend only linearly on the number of basis functions of the space discretization. With the help of the Fast Fourier Transform (FFT) algorithm, we can even extend our integrators to general potentials in a very natural way. In this case, too, the CPU effort grows only mildly as a function of the number of basis functions. In the physically relevant case of a fermionic system interacting via a contact interaction, we employ our newly developed schemes to conduct numerous simulations for temperatures closer and closer to the critical one. From these simulations, we conclude that the linearization deviates far from the original equations. More precisely, the linear approximation leads to an exponential decay of the Cooper pair density whereas the full equations yield oscillations about a finite value. Consequently, the diffusion which is inherent to all hitherto existing macroscopic theories can only be an unphysical artifact. With this, we add an important fact to the still ongoing discussion in the physics community. In short, we successfully developed convenient tools for the simulation of important physical phenomena in two fundamental fields of physics. This allowed our collaborators to gain valuable insights into the behavior of their equations of interest, thus contributing to the advance of fundamental science