A splitting method for the nonlinear Schrödinger equation

Abstract

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L 2 (R d )-theory.More precisely, we prove that the scheme is of first order in the L 2 (R d )-norm for H 2 (R d )-initial data.

Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation

“…See Proposition IV.6 of [6], where the calculus of Lie derivatives was used. Moreover, for nonlinearities of the type iλ|u| p u with p < 4/3 in [12] first order convergence of the Lie splitting in L 2 was shown for initial values in H 2 by different methods than ours. In this paper we focus on the time integration and do not treat the space discretization (which was studied in e.g.…”

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

“…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. Also, Ignat [12] introduced the following modified version of the splitting scheme:…”

“…In the following theorems, we provide the convergence results for initial data in H 1 pR d q. In order to obtain a convergence result with the low regularity assumption, Strichartz-type estimates are employed in [12] along with the Duhamel-type formula for Z τ , given by Z τ pnτ q " S τ pnτ qφ`τ n´1 ÿ k"0 S τ pnτ´kτ q N pτ q´I τ Z τ pkτ q, n ě 1, (1.4) which is similar to the Duhamel formula of the solution u to (1.1), expressed as uptq " Sptqφ`iλ ż t 0 Spt´sq|u| p upsqds, t ě 0.…”

“…In the above, the operator I in (1.4) is the identity operator, and the formula (1.4) is obtained as follows (refer to (2.11) in [12]):…”

High frequency wave packets for the Schrödinger equation and its numerical approximations

“…The theoretical results in [7] show that dispersive schemes ensure a polynomial convergence order which improves the logarithmic one gets for standard finitedifference schemes by energy methods. Also the effect of splitting methods [5,3] on these high frequency wave packets is worth investigating; iii) The issues addressed in this Note are totally open for non-uniform grids or Schrödinger equations in heterogeneous media.…”