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

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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. 
@article{ZVMMF_2011_51_3_a1,
     author = {I. A. Zlotnik},
     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},
     pages = {384--406},
     publisher = {mathdoc},
     volume = {51},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_3_a1/}
}
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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/