A new error analysis of a linearized Euler Galerkin scheme for Schrödinger equation with cubic nonlinearity

Abstract

In this paper, a linearized Euler Galerkin scheme is studied and the unconditionally optimal error estimate in $L^2$-norm is obtained for Schrödinger equation with cubic nonlinearity without any time-step restriction. The key to the analysis is to bound the $H^1$-norm between the numerical solution and the Ritz projection of the exact solution by mathematical induction for two cases rather than the error splitting technique used in the previous work. Finally, some numerical results are presented to confirm the theoretical analysis.