A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions

Abstract

This article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of O(h2+τ2)$$ O\left({h}^2+{\tau}^2\right) $$ with time step τ$$ \tau $$ and mesh size h$$ h $$ in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete L2$$ {L}^2 $$ norm or H1$$ {H}^1 $$ norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.