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 H4H^4 -regular solution, a first-order error bound in the H1H^1 norm is shown and used to derive a second-order error bound in the L2L_2 norm. For the cubic Schrödinger equation with an H4H^4 -regular solution, first-order convergence in the H2H^2 norm is used to obtain second-order convergence in the L2L_2 norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and HmH^m -conditional stability for error propagation, where m=1m=1 for the Schrödinger-Poisson system and m=2m=2 for the cubic Schrödinger equation.