A new error analysis of a linearized BDF2 Galerkin scheme for Schrödinger equation with cubic nonlinearity

Abstract

In this paper, a linearized 2-step backward differentiation formula (BDF2) Galerkin method is proposed and investigated for Schrödinger equation with cubic nonlinearity and unconditionally optimal error estimate in $L^2$-norm is obtained 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.