This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete H1 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| 2p, 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.
|Number of pages
|Japan Journal of Industrial and Applied Mathematics
|Published - 2016 Jul 1
- Finite difference method
- Nonlinear Schrödinger equation