Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation

This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete H 1 framework to establish well-posedness and error estimates in the L ∞ norm. The nonlinearity f ( u ) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity f ( u ) = - | u | 2 p , p being a positive real number. Particularly, we offer the numerical blow-up time T ( h , τ ) , where h and τ are discretization parameters of space and time variables. We prove that T ( h , τ ) converges to the blow-up time T ∞ of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of T ( h , τ ) is at a second order rate in τ if the Crank–Nicolson scheme is applied to time discretization.