High-order structure-preserving Du Fort-Frankel schemes and their analyses for the nonlinear Schrodinger equation with wave operator

作者：Deng, Dingwen^*; Li, Zhijun

来源：Journal of Computational and Applied Mathematics, 2023, 417: 114616.

DOI：10.1016/j.cam.2022.114616

• Du Fort-Frankel difference schemes
• Energy-and mass-conservation laws
• Convergence
• Nonlinear Schr?dinger equations with wave operator

Du Fort-Frankel-type finite difference methods (DFFT-FDMs) are famous for good sta-bility and easy implementation. In this study, by a perfect combination of the classical fourth-order difference to approximate the second-order spatial derivatives with the idea of DFFT-FDMs, a class of high-order structure-preserving DFFT-FDMs (SP-DFFT-FDMs) are firstly developed for solving the periodic initial-boundary value problems (PIBVPs) of one-dimensional (1D) and two-dimensional (2D) nonlinear Schrodinger equations with wave operator (NLSW), respectively. By using the discrete energy method, it is shown that their solutions satisfy the discrete energy-and mass-conservative laws, and are conditionally convergent with an order of O(& UTau;2 + h4x + (hx & UTau; )2) and O(& UTau;2 + h4x + h4y + ( hx & UTau; )2 + (hy & UTau; )2) in the discrete H1-norm, respectively. Here, & UTau; denotes time step size, while, hx and hy represent spatial mesh sizes in x and y directions, respectively. Then, by supplementing a stabilized term, a type of stabilized SP-DFFT-FDMs are devised. They not only preserve the discrete energy-and mass-conservation laws, but also own much better stability than original SP-DFFT-FDMs. Finally, numerical results confirm the theoretical findings, and the efficiency of our algorithms.

• 单位
  南昌航空大学