Optimal error estimate of a conservative, efficient and accurate finite difference scheme for the nonlinear Schrodinger equation with Dirac delta potentials

Abstract

This paper is concerned with constructing and analyzing an efficient and accurate finite difference scheme for the nonlinear Schrödinger equation with Dirac delta potentials. The proposed scheme exhibits several notable features: (1) It is derived from an accurate approximation of the internal interface matching conditions, enabling the use of varying mesh sizes in subintervals divided by the singular points ${\xi }_z(z=1,2,\cdots ,k)$. This ensures that all singular points ${\xi }_z(z=1,2,\cdots ,k)$ align with the grid nodes; (2) The scheme is proved to preserve the total mass and energy in the discrete sense; (3) The nonlinear term is discretized in a way that facilitates temporal linearization, and the spatial grid stencil comprises only three nodes. This translates to solving a tridiagonal system of linear algebraic equations efficiently using the Thomas algorithm at each time step; (4) The convergence order of the proposed scheme is proved to be $O(h^2+{\tau }^2)$ in the maximum norm, with no restrictions on the grid ratio, where, $h$ represents the mesh size and $\tau$ denotes the time step. We then derive two other efficient and accurate finite difference schemes by enhancing the accuracy of the approximation of the internal interface matching conditions, one still preserves the mass and energy in the discrete sense but needs uniform grid, the other one is nonconservative but allows different mesh sizes in different subintervals. Numerical results are carried out to validate our theoretical conclusions and simulate several dynamical behaviors of the nonlinear Schrödinger equation with Dirac delta potentials.