Implicit finite-difference schemes, based on the Rosenbrock method, for nonlinear Schrödinger equation with artificial boundary conditions.

Vyacheslav A Trofimov,Evgeny M Trykin,Chih-Chun Chien

doi:10.1371/journal.pone.0206235

Vyacheslav A Trofimov, Evgeny M Trykin + Show 1 more

Open Access

https://doi.org/10.1371/journal.pone.0206235

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Journal: PloS one	Publication Date: Oct 31, 2018
Citations: 7	License type: CC BY 4.0

Affiliation: Lomonosov Moscow State University

Abstract

We investigate the effectiveness of using the Rosenbrock method for numerical solution of 1D nonlinear Schrödinger equation (or the set of equations) with artificial boundary conditions (ABCs). We compare the computer simulation results obtained during long time interval at using the finite-difference scheme based on the Rosenbrock method and at using the conservative finite-difference scheme. We show, that the finite-difference scheme based on the Rosenbrock method is conditionally conservative one. To combine the advantages of both numerical methods, we propose new implicit and conditionally conservative combined method based on using both the conservative finite-difference scheme and conditionally conservative Rosenbrock method and investigate its effectiveness. The combined method allows decreasing the computer simulation time in comparison with the corresponding computer simulation time at using the Rosenbrock method. In practice, the combined method is effective at computation during short time interval, which does not require an asymptotic stability property for the finite-difference scheme. We generalize also the combined method with ABCs for 2D case.

Highlights

As is well-known, wide physical phenomena (starting from laser radiation propagation in a nonlinear medium up to quantum mechanics problems and Bose-Einstein condensate (BEC)) are described by the nonlinear Schrodinger equation
We demonstrate this feature in Table 2: for the time step τ = 0.001 the first invariant and Hamiltonian deviations occurring at using the Rosenbrock methods and the conservative finite-difference schemes (CFDS) have the similar order of magnitude and numerical solutions for all methods coincide each other
In this paper we have shown the conditional conservatism property of the finite-difference scheme based on the Rosenbrock method for the 1D nonlinear Schrodinger equation or the set of equations

Summary

Introduction

As is well-known, wide physical phenomena (starting from laser radiation propagation in a nonlinear medium up to quantum mechanics problems and Bose-Einstein condensate (BEC)) are described by the nonlinear Schrodinger equation (or set of the equations). In [9] it is shown the significant changing (about unity) of the third invariant (the Hamiltonian) of the problem at computation on the base of the split-step method These papers do not contain the detailed comparison between the finite-difference scheme based on the Rosenbrock method and the CFDS, developed for the nonlinear Schrodinger equation. That in our opinion the one-stage Rosenbrock scheme with parameter b 1⁄4 1 corresponds to the Eq (12) right part approximation by the half-sum of the mesh functions from the upper and current time layers (so called, the Crank-Nikolson scheme) This type of the approximation is preferable for the nonlinear Schrodinger equation with a cubic nonlinear response because it allows us to achieve the conservative property with respect to the Hamiltonian (the third invariant (7)). Fj is a vector of dimension 2, which has the form:

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