Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

Abstract

Abstract. In this paper, we study the linear Schrödinger equation over the d-dimensional torus,with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are exponentially long with the time discretization parameter. We here refer to Gevrey regularity, and our estimates turn out to be uniform in the space discretization parameter. This paper extends [G. Dujardin and E. Faou, Numer. Math. 97 (2004) 493-535], where a similar result has been obtained in the semi-discrete situation, i.e. when the mere time variable is discretized and space is kept a continuous variable.Mathematics Subject Classification. 65P10, 37M15, 37K55.

Splitting Integrators for Nonlinear Schrödinger Equations Over Long Times

“…Here, we extend these results to full discretizations using in addition a LieTrotter splitting in time. A long-time analysis of semi-discretizations in time via a Lie-Trotter splitting for linear Schrödinger equations with a multiplicative potential has been performed by Dujardin and Faou [6] and was recently transferred to a full discretization by Castella and Dujardin [3]. Splitting integrators for a general class of infinite-dimensional Hamitonian systems including the nonlinear Schrödinger equation have very recently been studied independently and with a different approach by Faou, Grébert, and Paturel [8] and [9].…”

“…The considered discretization of (1) is presented in Sect. 3. In the subsequent sections we analyze the (almost-) invariants of the exact solution along this numerical approximation.…”

Numerical Analysis of Schrödinger Equations in the Highly Oscillatory Regime

“…Here the energy band E m (k) is understood to be periodically extended to all of R. To this end, note that the following relation holds 16) as can be shown by a lengthy but straightforward calculation. Of course if U ≡ 0 (the non-periodic part of the potential) the time evolution (5.1) in general mixes all band spaces H m , i.e.…”

“…In the second part of the paper we consider time splitting-trigonometric spectral schemes which have much better asymptotic properties as → 0. For analytical results on time-splitting spectral methods for linear and nonlinear Schrödinger equation (not in the semiclassical regime, though) we refer to [16], [28], [24]. The third part of the paper is concerned with an extension of the spectral-time splitting scheme to Schrödinger equations with periodic highly oscillatory potentials, typically occuring in solid state physics.…”

On energy conservation by trigonometric integrators in the linear case with application to wave equations

“…The term g(q) = −∇U (q) in ( 1) stems from a (sufficiently regular, in particular real differentiable) potential U : C d → R. The complex gradient ∇ with respect to q ∈ C d is defined as ∇ = ∇ x + i∇ y with the real part x ∈ R d and the imaginary part y ∈ R d of q = x + iy. 1 We will be interested in the case that ( 1) is a linear equation, i.e., g(q) = −Aq,…”