@Article{CiCP-29-5,
author = {Marco, Caliari 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 = {<p style="text-align: justify;">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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0131},
url = {https://global-sci.com/article/79659/a-fast-time-splitting-finite-difference-approach-to-grosspitaevskii-equations}
}