Geodesic
Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 3, pp. 384-406 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary {Schr\"odinger} equation in a~semi-infinite strip},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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I. A. Zlotnik. Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 3, pp. 384-406. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_3_a1/

[1] Berger J.-F., Girod M., Gogny D., “Time-dipendent quantum collective dynamics applied to nuclear fission”, Comp. Phys. Communs., 63 (1991), 365–374 | DOI | Zbl

[2] Goutte H., Berger J.-F., Casoly P., Gogny D., “Microscopic approach of fission dynamics applied to fragment kinetic energy and mass distribution in ${}^{238}$U”, Phys. Rev. C, 71:2 (2005), 024316, 13 pp. | DOI

[3] Antoine X., Arnold A., Besse C., Ehrhardt M., Schadle A., “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations”, Commun. Compus. Phys., 4:4 (2008), 729–796 | MR

[4] Arnold A., “Numerically absorbing boundary conditions for quantum evolution equations”, VLSI Design, 6 (1998), 313–319 | DOI

[5] Ehrhardt M., Arnold A., “Discrete transparent boundary conditions for the Schrödinger equation”, Riv. Mat. Univ. Parma, 6 (2001), 57–108 | MR | Zbl

[6] Arnold A., Ehrhardt M., Sofronov I., “Disctrete transparent boundary conditions for the Schödinger equation: fast calculations, approximation and stability”, Communs. Math. Sci., 1 (2003), 501–556 | MR | Zbl

[7] Ducomet B., Zlotnik A., “On stability of the Crank–Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part I”, Communs. Math. Sci., 4:4 (2006), 741–766 | MR | Zbl

[8] Ducomet B., Zlotnik A., “On stability of the Crank–Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part II”, Communs. Math. Sci., 5:2 (2007), 267–298 | MR | Zbl

[9] Zlotnik A. A., Dyukome B., “Ustoichivost simmetrichnoi raznostnoi skhemy s priblizhennymi prozrachnymi granichnymi usloviyami dlya nestatsionarnogo uravneniya Shredingera”, Dokl. RAN, 413:134 (2007), 96–107 | MR

[10] Schmidt F., Yevick D., “Discrete transparent boundary conditions for Schrödinger-type equations”, J. Comput. Phys., 134 (1997), 96–107 | DOI | MR | Zbl

[11] Antoine X., Besse C., “Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation”, J. Compus. Phys., 188 (2003), 157–175 | DOI | MR | Zbl

[12] Han H., Jin J., Wu X., “A finite-difference method for the one-dimensional time-dependent Schrödinger-type equation on unbounded domains”, Comput. Math. Appl., 50 (2005), 1345–1362 | DOI | MR | Zbl

[13] Moyer C. A., “Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension”, Amer. J. Phys., 72:3 (2004), 351–358 | DOI

[14] Schulte M., Arnold A., “Discrete transparent boundary conditions for the Schrödinger equation, a compact higher order scheme”, Kinetic and Related Models, 1:1 (2008), 101–125 | MR | Zbl

[15] Zlotnik A. A., Zlotnik I. A., “Ob ustoichivosti semeistva raznostnykh skhem s priblizhennymi prozrachnymi granichnymi usloviyami dlya uravneniya Shredingera na poluosi”, Vestn. MEI, 2008, no. 6, 31–45

[16] Ducomet B., Zlotnik A., Zlotnik I., “On a family of finite-difference schemes with discrete transparent boundary conditions for a generalized Schrödinger equation”, Kinetic and Related Models, 2:1 (2009), 151–180 | DOI | MR

[17] Zlotnik I. A., “Ob ustoichivosti semeistva raznostnykh skhem s priblizhennymi prozrachnymi granichnymi usloviyami dlya nestatsionarnogo uravneniya Shredingera v polupolose”, Vestn. MEI, 2009, no. 6, 127–144

[18] Zlotnik A. A., “Some finite-element and finite-difference methods for solving mathematical physics problems with non-smooth data in n-dimensional cube”, Sovjet. J. Numer. Anal. Math. Modelling, 6:5 (1991), 421–451 | DOI | MR | Zbl

[19] Zlotnik I. A., “Kompyuternoe modelirovanie tunnelnogo effekta”, Vestn. MEI, 2010, no. 6, 10–28