Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations with Wave Operator

Du Fort-Frankel finite difference method (FDM) was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is an explicit and unconditionally von Neumann stable scheme. However, there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods (FDMs). In this study, a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional (1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator. By using the discrete energy method, it is shown that their solutions possess the discrete energy and mass conservative laws, and conditionally converge to exact solutions with an order of $\mathcal{O}(τ^2+h^2_x+(τ/h_x)^2)$ for 1D problem and an order of $\mathcal{O}(τ^2+h^2_x+h^2_y+(τ/h_x)^2+(τ/h_y)^2)$ for 2D problem in $H^1$-norm. Here, $τ$ denotes time-step size, while, $h_x$ and $h_y$ represent spatial meshsizes in $x$- and $y$-directions, respectively. Then, by introducing a stabilized term, a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised. They not only preserve the discrete energies and masses, but also own much better stability than original schemes. Finally, numerical results demonstrate the theoretical analyses.