A simple Dufort-Frankel type scheme for the Gross-Pitaevskii equation of Bose-Einstein condensates on different geometries

Abstract

We develop a simple Dufort-Frankel type scheme for solving the time-dependent Gross-Pitaevskii equation (GPE). The GPE is a nonlinear Schrodinger equation describing the Bose-Einstein condensation (BEC) at very low temperature. Three different geometries including 1D spherically symmetric, 2D cylindrically symmetric, and 3D anisotropic Cartesian domains are considered. The present finite difference method is explicit, linearly unconditional stable and is able to han- dle the coordinate singularities in a natural way. Furthermore, the scheme is time reversible and satisfies a discrete analogue of density conservation law.