Error Estimation of the Relaxation Finite Difference Scheme for the Nonlinear Schrödinger Equation

Abstract

We consider an initial- and boundary-value problem for the nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference method and in time by the Relaxation Scheme proposed by C. Besse [C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1427–1432]. We provide optimal order error estimates, in the discrete -norm, for the approximation error at the time nodes and at the intermediate time nodes. In the context of the nonlinear Schrödinger equation, this is the first time that the derivation of an error estimate, for a fully discrete method based on the Relaxation Scheme, is completely addressed.