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

• Main Authors: Jürgen Geiser, Amirbahador Nasari
• Format: Article
• Language: English
• Published: MDPI AG 2019-08-01
• Series: Mathematical and Computational Applications
• Subjects: nonlinear Schrödinger equation; Gross–Pitaevskii equation; Bose–Einstein condensates; Spinor systems; splitting methods; splitting spectral methods; convergence analysis; conservation methods
• Online Access: https://www.mdpi.com/2297-8747/24/3/76

In this paper, we discuss the different splitting approaches to numerically solve<br />the Gross–Pitaevskii equation (GPE). The models are motivated from spinor Bose–Einstein<br />condensate (BEC). This system is formed of coupled mean-field equations, which are based on<br />coupled Gross–Pitaevskii equations. We consider conservative finite-difference schemes and spectral<br />methods for the spatial discretisation. Furthermore, we apply implicit or explicit time-integrators and<br />combine these schemes with different splitting approaches. The numerical solutions are compared<br />based on the conservation of the <em>L</em>₂-norm with the analytical solutions. The advantages of the novel<br />splitting methods for large time-domains are based on the asymptotic conservation of the solution of<br />the soliton’s applications. Furthermore, we have the benefit of larger local time-steps and therefore<br />obtain faster numerical schemes.