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

Abstract

In 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.

Superconvergence of time invariants for the Gross-Pitaevskii equation

“…Crucial for the proof is thus the following semi-discrete auxiliary problem whose properties have been studied in [31] and [33]. Lemma 9.3 (semi-discrete Crank-Nicolson scheme).…”

“…A proof of the L ∞ (H 2 ) and L ∞ (L 2 ) estimates is given in [31] a proof of the L ∞ (H 1 ) estimate is given in [33]. As we will see later, the L ∞ (H 2 )-estimate is not optimal and can be improved by one order.…”

“…Practically, the discrete conservation of mass and energy is subject to the choice of the time integrator. Among others, mass conservative time discretizations have been studied in [57,61], time integrators that are mass conservative and symplectic are investigated in [3,25,51,54,56], energy conservative time discretizations in [34] and time discretization that preserve mass and energy simultaneously are addressed in [3,7,9,11,12,16,31,33,50,62]. For further discretizations we refer to [5,8,10,36,53] and the references therein.…”

Uniform $L^\infty$-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

“…Time integrators that conserve a modified energy are for example the popular Besse relaxation scheme [15,16,52] and the family of exponential Runge-Kutta schemes proposed in [26]. Time integrators that conserve the exact energy (up to spatial discretization errors) are more rare and include the Crank-Nicolson method based on the averaging of densities [3,10,32,35,45] and the continuous Galerkin time stepping proposed in [39]. The latter method is also the numerical scheme that we shall consider in this paper, as it is not only energy-conservative, but it also allows for arbitrarily fast convergence rates for smooth solutions.…”

“…The first application of this technique to nonlinear Schrödinger equations such as the GPE was established by Wang [49] who considered an Adams-Bashforth-type linearization of the Crank-Nicolson method. Applications to the energyconservative Crank-Nicolson method were developed in [32,35]. In all cases, lowest order finite element spaces were considered.…”