Error estimate of the conservative difference scheme for the derivative nonlinear Schrödinger equation

Abstract

In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order $O({\tau }^2+h^2)$ under discrete $L^{\infty }$-norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.