Optimal Point-Wise Error Estimate of a Compact Finite Difference Scheme for the Coupled Nonlinear Schrödinger Equations (Journal of Computational Mathematics, 2018)
In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear Schrödinger equation. The proposed scheme not only conserves the total mass and energy in the discrete level but also is decoupled and linearized in practical computation. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of $O(h^4+τ^2)$ in the discrete $L^∞$-norm with time step $τ$ and mesh size $h$. Finally, numerical results are reported to test our theoretical results of the proposed scheme.
Keywords: Coupled nonlinear Schrödinger equations; Compact difference scheme; Conservation; Point-wise error estimate.