A new linear and conservative finite difference scheme for the Gross–Pitaevskii equation with angular momentum rotation

Abstract

A new linear and conservative finite difference scheme which preserves discrete mass and energy is developed for the two-dimensional Gross–Pitaevskii equation with angular momentum rotation. In addition to the energy estimate method and mathematical induction, we use the lifting technique as well as some well-known inequalities to establish the optimal \(H^{1}\)-error estimate for the proposed scheme with no restrictions on the grid ratio. Unlike the existing numerical solutions which are of second-order accuracy at the most, the convergence rate of the numerical solution is proved to be of order \(O(h^4+\tau^2)\) with time step \(\tau\) and mesh size \(h\). Numerical experiments have been carried out to show the efficiency and accuracy of our new method. doi:10.1017/S1446181119000026