Efficient time integration methods for Gross-Pitaevskii equations with rotation term

Philipp Bader; Sergio Blanes; Fernando Casas; Mechthild Thalhammer

doi:10.3934/jcd.2019008

Article Contents

2019, Volume 6, Issue 2: 147-169. Doi: 10.3934/jcd.2019008

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Efficient time integration methods for Gross-Pitaevskii equations with rotation term

1.
Universitat Jaume I, Departament de Matemàtiques, 12071 Castellón, Spain
2.
Universitat Politècnica de València, Instituto Universitario de Matemática Multidisciplinar, 46022 Valencia, Spain
3.
Universitat Jaume I, IMAC and Departament de Matemàtiques, 12071 Castellón, Spain
4.
Leopold-Franzens-Universität Innsbruck, Institut für Mathematik, 6020 Innsbruck, Austria

Received: February 28, 2019

Revised: July 31, 2019

Published: November 2019

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Abstract

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.

Keywords:

Nonlinear Schrödinger equations,
Gross-Pitaevskii equations,
exponential integrators,
Magnus integrators,
commutator-free quasi-Magnus integrators,
spectral methods,
fast Fourier transform.

Mathematics Subject Classification: Primary: 35Q55, 65L20, 65M12; Secondary: 58G35.

Citation:

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Figure 1. Overview of exponential time integration methods. Commutator-free quasi-Magnus exponential integrators (CFQM, order $ p $, number of compositions $ J $, number of quadrature nodes $ K $) or modified commutator-free quasi-Magnus exponential integrator (Scheme BBK, order $ p = 6 $), respectively, combined with operator splitting methods (order $ p $, number of compositions $ s $)

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Figure 2. Time integration of non-autonomous linear test equation in two (left) and three (right) space dimensions by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively

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Figure 3. Time integration of non-autonomous nonlinear test equation in two (left) and three (right) space dimensions with $ \vartheta = 1 $ by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively

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Figure 4. Time integration of non-autonomous nonlinear test equation in two (left) and three (right) space dimensions with $ \vartheta = 10 $ by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively

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Figure 5. Time integration of two-dimensional rotational Gross-Pitaevskii equation (22) by sixth-order modified CFQM exponential integrator (20) and optimised sixth-order splitting method proposed in [18]. Solution profile $ |{\psi(x, t)}|^2 $ displayed for spatial section $ x \in [- 5, 5]^2 $ and time $ t = 15 $. Consistent results obtained by time-splitting generalised-Laguerre-Fourier spectral methods [22,Eq. (7)-(8),Fig. 2]

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