Open AccessImplicit finite-difference schemes, based on the Rosenbrock method, for nonlinear Schrödinger equation with artificial boundary conditions
Author(s) -
Vyacheslav A. Trofimov,
Evgeny M. Trykin
Publication year - 2018
Publication title -
plos one
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.99
H-Index - 332
ISSN - 1932-6203
DOI - 10.1371/journal.pone.0206235
Subject(s) - finite difference method , nonlinear system , interval (graph theory) , mathematics , computation , finite difference , scheme (mathematics) , boundary value problem , stability (learning theory) , boundary (topology) , numerical analysis , set (abstract data type) , computer science , algorithm , mathematical analysis , physics , combinatorics , quantum mechanics , machine learning , programming language
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.