Optimal Point-Wise Error Estimate of a Compact Difference Scheme for the Coupled Gross–Pitaevskii Equations in One Dimension

The coupled Gross–Pitaevskii (CGP) equation studied in this paper is an important mathematical model describing two-component Bose–Einstein condensate with an internal atomic Josephson junction. We here analyze a compact finite difference scheme which conserves the total mass and energy in the discrete level for the CGP equation. In general, due to the difficulty caused by compact difference on nonlinear terms, optimal point-wise error estimates without any restrictions on the grid ratios of compact difference schemes for nonlinear partial differential equations are very hard to be established. To overcome the difficulty caused by the compact difference operator, we introduce a new norm and an interesting transformation by which the difference scheme is transformed into a special equivalent vector form, we then use the energy method and some important lemmas on the equivalent system to obtain the optimal convergent rate, without any restrictions on the grid ratio, at the order of $$O(h^{4}+\tau ^2)$$O(h4+τ2) in the maximum norm with time step $$\tau $$τ and mesh size $$h$$h. Finally, numerical results are reported to test the theoretical results and simulate the dynamics of the CGP equation.