Volume 29, Issue 5
A Fast Time Splitting Finite Difference Approach to Gross–Pitaevskii Equations

Commun. Comput. Phys., 29 (2021), pp. 1336-1364.

Published online: 2021-03

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• Abstract

We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced 4th-order finite differences. This results in a free boundary approach, which can be solved in time by the Strang splitting method. The linear part is solved by a new, fast approximation of the action of the matrix exponential at machine precision accuracy, while the nonlinear part can be solved exactly. Numerical results confirm existing ones based on the Fourier pseudospectral method and point out some weaknesses of the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the vortical structures, of their reconnections, and of integral quantities like mass, energy, and momentum. The free boundary approach reproduces them correctly, also in finite subdomains, at low computational cost. We show the versatility of this method by carrying out one- and three-dimensional simulations and by using it also in the case of Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity.

• Keywords

Gross–Pitaevskii boundary conditions, dark structures, vortex rings, unbounded domain, nonlinear Schrödinger equation, fast matrix exponential.

65M06,65M70

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@Article{CiCP-29-1336, author = {Caliari , Marco and Zuccher , Simone}, title = {A Fast Time Splitting Finite Difference Approach to Gross–Pitaevskii Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {5}, pages = {1336--1364}, abstract = {

We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced 4th-order finite differences. This results in a free boundary approach, which can be solved in time by the Strang splitting method. The linear part is solved by a new, fast approximation of the action of the matrix exponential at machine precision accuracy, while the nonlinear part can be solved exactly. Numerical results confirm existing ones based on the Fourier pseudospectral method and point out some weaknesses of the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the vortical structures, of their reconnections, and of integral quantities like mass, energy, and momentum. The free boundary approach reproduces them correctly, also in finite subdomains, at low computational cost. We show the versatility of this method by carrying out one- and three-dimensional simulations and by using it also in the case of Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0131}, url = {http://global-sci.org/intro/article_detail/cicp/18716.html} }
TY - JOUR T1 - A Fast Time Splitting Finite Difference Approach to Gross–Pitaevskii Equations AU - Caliari , Marco AU - Zuccher , Simone JO - Communications in Computational Physics VL - 5 SP - 1336 EP - 1364 PY - 2021 DA - 2021/03 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0131 UR - https://global-sci.org/intro/article_detail/cicp/18716.html KW - Gross–Pitaevskii boundary conditions, dark structures, vortex rings, unbounded domain, nonlinear Schrödinger equation, fast matrix exponential. AB -

We propose an idea to solve the Gross–Pitaevskii equation for dark structures inside an infinite constant background density $ρ_∞$=${|ψ_∞|}^2$, without the introduction of artificial boundary conditions. We map the unbounded physical domain $\mathbb{R}^3$ into the bounded domain ${(−1,1)}^3$ and discretize the rescaled equation by equispaced 4th-order finite differences. This results in a free boundary approach, which can be solved in time by the Strang splitting method. The linear part is solved by a new, fast approximation of the action of the matrix exponential at machine precision accuracy, while the nonlinear part can be solved exactly. Numerical results confirm existing ones based on the Fourier pseudospectral method and point out some weaknesses of the latter such as the need of a quite large computational domain, and thus a consequent critical computational effort, in order to provide reliable time evolution of the vortical structures, of their reconnections, and of integral quantities like mass, energy, and momentum. The free boundary approach reproduces them correctly, also in finite subdomains, at low computational cost. We show the versatility of this method by carrying out one- and three-dimensional simulations and by using it also in the case of Bose–Einstein condensates, for which $ψ$→0 as the spatial variables tend to infinity.

Marco Caliari & Simone Zuccher. (2021). A Fast Time Splitting Finite Difference Approach to Gross–Pitaevskii Equations. Communications in Computational Physics. 29 (5). 1336-1364. doi:10.4208/cicp.OA-2020-0131
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