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Open AccessFeature PaperArticle

Comparison of Splitting Methods for Deterministic/Stochastic Gross–Pitaevskii Equation

1
The Institute of Theoretical Electrical Engineering, Ruhr University of Bochum, Universitätsstrasse 150, D-44801 Bochum, Germany
2
Department of Civil and Environmental Engineering, Ruhr University of Bochum, Universitätsstrasse 150, D-44801 Bochum, Germany
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(3), 76; https://doi.org/10.3390/mca24030076
Received: 12 March 2019 / Revised: 16 August 2019 / Accepted: 18 August 2019 / Published: 20 August 2019
(This article belongs to the Section Engineering)
In this paper, we discuss the different splitting approaches to numerically solve the Gross–Pitaevskii equation (GPE). The models are motivated from spinor Bose–Einstein condensate (BEC). This system is formed of coupled mean-field equations, which are based on coupled Gross–Pitaevskii equations. We consider conservative finite-difference schemes and spectral methods for the spatial discretisation. Furthermore, we apply implicit or explicit time-integrators and combine these schemes with different splitting approaches. The numerical solutions are compared based on the conservation of the L 2 -norm with the analytical solutions. The advantages of the novel splitting methods for large time-domains are based on the asymptotic conservation of the solution of the soliton’s applications. Furthermore, we have the benefit of larger local time-steps and therefore obtain faster numerical schemes. View Full-Text
Keywords: nonlinear Schrödinger equation; Gross–Pitaevskii equation; Bose–Einstein condensates; Spinor systems; splitting methods; splitting spectral methods; convergence analysis; conservation methods nonlinear Schrödinger equation; Gross–Pitaevskii equation; Bose–Einstein condensates; Spinor systems; splitting methods; splitting spectral methods; convergence analysis; conservation methods
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Geiser, J.; Nasari, A. Comparison of Splitting Methods for Deterministic/Stochastic Gross–Pitaevskii Equation. Math. Comput. Appl. 2019, 24, 76.

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