Optimal l∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate, it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions (2D or 3D). We here propose and analyze finite difference methods for solving the coupled Gross-Pitaevskii equations in two dimensions, which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction. The methods which we considered include two conservative type schemes and two non-conservative type schemes. Discrete conservation laws and solvability of the schemes are analyzed. For the four proposed finite difference methods, we establish the optimal convergence rates for the error at the order of O(h2 +τ2) in the l∞-norm (i.e., the point-wise error estimates) with the time step τ and the mesh size h. Besides the standard techniques of the energy method, the key techniques in the analysis is to use the cut-off function technique, transformation between the time and space direction and the method of order reduction. All the methods and results here are also valid and can be easily extended to the three-dimensional case. Finally, numerical results are reported to confirm our theoretical error estimates for the numerical methods.