An efficient and conservative compact finite difference scheme for the coupled Gross–Pitaevskii equations describing spin-1 Bose–Einstein condensate

Abstract

The coupled Gross–Pitaevskii system studied in this paper is an important mathematical model describing spin-1 Bose-Einstein condensate. We propose a linearized and decoupled compact finite difference scheme for the coupled Gross–Pitaevskii system, which means that only three tri-diagonal systems of linear algebraic equations at each time step need to be solved by using Thomas algorithm. New types of mass functional, magnetization functional and energy functional are defined by using a recursive relation to prove that the new scheme preserves the total mass, energy and magnetization in the discrete sense. Besides the standard energy method, we introduce an induction argument as well as a lifting technique to establish the optimal error estimate of the numerical solution without imposing any constraints on the grid ratios. The convergence order of the new scheme is of O(h4+τ2) in the L2 norm and H1 norm, respectively, with time step τ and mesh size h. Our analysis method can be used to high dimensional cases and other linearized finite difference schemes for the two- or three-dimensional nonlinear Schrödinger/Gross–Pitaevskii equations. Finally, numerical results are reported to test the theoretical results.