A note on optimal $$H^1$$-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation

Abstract

AbstractIn this paper we consider a mass- and energy–conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal $$L^{\infty }(H^1)$$ L ∞ ( H 1 ) -error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the arising nonlinear system of equations that makes the method easy to implement and efficient. This is illustrated by numerical experiments.