Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation

Abstract

Abstract. We investigate the Lie and the Strang splitting for the cubic nonlinear Schrödinger equation on the full space and on the torus in up to three spatial dimensions. We prove that the Strang splitting converges in L 2 with order 1 + θ for initial values in H 2+2θ with θ ∈ (0, 1) and that the Lie splitting converges with order one for initial values in H 2 .

Low Regularity Exponential-Type Integrators for Semilinear Schrödinger Equations

“…The experiments show that Strang splitting is second-order convergent for ϑ ≥ 5, and that Lie splitting is first-order convergent for ϑ ≥ 3. These observations are in line with the convergence proofs from the literature [10,25], see also our discussion in the introduction. For smaller values of ϑ, both Lie and Strang splitting show a zig-zag behaviour, depending on whether the local errors accumulate or happy error cancellation occurs.…”

“…Due to the appearance of ∆u in the local error, the boundedness of at least two additional derivatives of the exact solution is required. Based on [25], fractional error estimates for the Strang splitting were established in [10] which require the boundedness of 2 + 2γ additional derivatives for convergence of order 1 + γ, with 0 < γ < 1. Furthermore, in [19] first-order convergence of a filtered Lie splitting method for Schrödinger equations of type (1) with 1 ≤ 2p < 4/d was shown in L 2 (R d ) for solutions in H 2 (R d ).…”

On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $

“…The convergence of these two schemes has been studied by Besse et al [2] for globally Lipschitz continuous nonlinearities and by Lubich [18] for Schrödinger-Poisson and cubic NLS equations with initial data in the space H 4 pR 3 q. Recently, Eilinghoff et al [5] established the convergence result for H 2`2 pR d q with P r0, 1q and 1 ď d ď 3. On the other hand, Ignat and Zuazua [13,14] and Ignat [11] developed various numerical schemes for which they proved Strichartz type estimates to obtain the convergence of the schemes with initial data of low regularity.…”

“…‚ (Eilinghoff-Schaubelt-Schratz [5]) Let 1 ď d ď 3 and p " 2. For the Strang approximation Z, we assume that φ P H 2`2 pR d q with P p0, 1q and that T ą 0 is a positive such that sup 0ďtďT }uptq} H 2`2 pR d q ă 8.…”

Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

“…Many numerical methods have been proposed for the integration of NSE, such as splitting methods (see, e.g., [13–21]), exponential‐type integrators (see, e.g., [11, 22–26]), multi‐symplectic methods (see, e.g., [27, 28]), and other effective methods (see, e.g., [29–33]). In recent decades, structure‐preserving (SP) algorithms of differential equations have been received much attention, and for the related work we refer the reader to [28, 34–44] and references therein.…”