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 Zh. Vychisl. Mat. Mat. Fiz., 2011, Volume 51, Number 3, Pages 384–406 (Mi zvmmf8070)

Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip

I. A. Zlotnik

Department of Mathematical Modeling, Moscow Power Engineering Institute (Technical University), Krasnokazarmennaya ul. 14, Moscow, 111250 Russia

Abstract: An initial–boundary value problem for the generalized Schrödinger equation in a semi-infinite strip is solved. A new family of two-level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed, and numerical results are presented.

Key words: nonstationary two-dimensional Schrödinger equation in unbounded domain, two-evel finite-difference schemes, approximate and discrete transparent boundary conditions, stability, finite-difference schemes, Matlab.

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English version:
Computational Mathematics and Mathematical Physics, 2011, 51:3, 355–376

Bibliographic databases:

UDC: 519.633

Citation: I. A. Zlotnik, “Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip”, Zh. Vychisl. Mat. Mat. Fiz., 51:3 (2011), 384–406; Comput. Math. Math. Phys., 51:3 (2011), 355–376

Citation in format AMSBIB
\Bibitem{Zlo11} \by I.~A.~Zlotnik \paper Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schr\"odinger equation in a~semi-infinite strip \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2011 \vol 51 \issue 3 \pages 384--406 \mathnet{http://mi.mathnet.ru/zvmmf8070} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2839572} \transl \jour Comput. Math. Math. Phys. \yr 2011 \vol 51 \issue 3 \pages 355--376 \crossref{https://doi.org/10.1134/S0965542511030122} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000289167800002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79953719420} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Degtyarëv A.A., Kozlova E.S., “Issledovanie pogreshnosti raznostnogo resheniya odnonapravlennogo uravneniya Gelmgoltsa metodom vychislitelnogo eksperimenta”, Kompyuternaya optika, 36:1 (2012), 36–45
2. Zlotnik A.A., Zlotnik I.A., “Finite element method with discrete transparent boundary conditions for the one-dimensional nonstationary Schrödinger equation”, Dokl. Math., 86:3 (2012), 750–755
3. Zlotnik A., Zlotnik I., “Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation”, Kinet. Relat. Models, 5:3 (2012), 639–667
4. Morandi O., “Mathematical Analysis of a Nonparabolic Two-Band Schrodinger-Poisson Problem”, Transport. Theor. Statist. Phys., 42:4-5 (2013), 133–161
5. Morandi O., “Existence of Solution of a Non-Linear Multiband KP Model With Transparent Boundary Conditions”, J. Phys. A-Math. Theor., 47:48 (2014), 485301
6. Ducomet B., Zlotnik A., Zlotnik I., “The Splitting in Potential Crank-Nicolson Scheme With Discrete Transparent Boundary Conditions For the Schrodinger Equation on a Semi-Infinite Strip”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 48:6 (2014), 1681–1699
7. Zlotnik A., Romanova A., “on a Numerov-Crank-Nicolson-Strang Scheme With Discrete Transparent Boundary Conditions For the Schrodinger Equation on a Semi-Infinite Strip”, Appl. Numer. Math., 93:SI (2015), 279–294
8. Ducomet B., Zlotnik A., Romanova A., “on a Splitting Higher-Order Scheme With Discrete Transparent Boundary Conditions For the Schrodinger Equation in a Semi-Infinite Parallelepiped”, Appl. Math. Comput., 255 (2015), 196–206
9. Zlotnik A., “Error Estimates of the Crank-Nicolson-Polylinear Fem With the Discrete Tbc For the Generalized Schrodinger Equation in An Unbounded Parallelepiped”, Finite Difference Methods, Theory and Applications, Lecture Notes in Computer Science, 9045, eds. Dimov I., Farago I., Vulkov L., Springer-Verlag Berlin, 2015, 129–141
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