On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

Abstract

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an H 4-regular solution, a first-order error bound in the H 1 norm is shown and used to derive a second-order error bound in the L 2 norm. For the cubic Schrödinger equation with an H 4-regular solution, first-order convergence in the H 2 norm is used to obtain second-order convergence in the L 2 norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and H m-conditional stability for error propagation, where m = 1 for the Schrödinger-Poisson system and m = 2 for the cubic Schrödinger equation.