Optimal point-wise error estimates of two conservative finite difference schemes for the coupled Gross–Pitaevskii equations with angular momentum rotation terms

Abstract

This paper is concerned with numerical analysis of two finite difference schemes for solving the coupled Gross–Pitaevskii equations with angular momentum rotation terms which model the dynamics of rotating two-component Bose–Einstein condensates. Both schemes are proved to be uniquely solvable and inherit the conservation laws of mass and energy in the discrete sense. Besides the energy method, the ‘cut-off’ function technique and inductive argument are introduced to establish the optimal error estimates of the numerical methods without any requirement on the grid-ratio or initial values. The convergence order of the numerical solutions to the exact solutions is proved to be of O(τ2+h2) with time-step τ and mesh size h. Several numerical results are reported to validate the error estimates and conservation laws.