Optimal error estimates of explicit finite difference schemes for the coupled Gross-Pitaevskii equations

In this paper, we investigate the convergence of explicit finite difference schemes which contain a Richardson scheme and a leap-frog scheme for computing the coupled Gross-Pitaevskii equations in high space dimensions. We establish the optimal error estimates of our schemes at the order of O(t 2 + h2) in the l8-norm with the time step t and the mesh size h. Besides the standard techniques of the energy analysis method, the key techniques in the analysis is to use the method of induction argument and order reduction. The numerical results are reported to confirm our theoretical analysis for the numerical methods.