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Finite element method with discrete transparent boundary conditions for the timedependent 1D Schrödinger equation
1.  Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russian Federation 
2.  Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation 
References:
[1] 
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comp. Phys., 4 (2008), 729796. 
[2] 
X. Antoine and C. Besse, Unconditionally stable discretization schemes of nonreflecting boundary conditions for the onedimensional Schrödinger equation, J. Comp. Phys., 188 (2003), 157175. doi: 10.1016/S00219991(03)001591. 
[3] 
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), 313319. 
[4] 
A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: Fast calculations, approximation, and stability, Comm. Math. Sci., 1 (2003), 501556. 
[5] 
B. Ducomet and A. Zlotnik, On stability of the CrankNicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Comm. Math. Sci., 4 (2006), 741766. 
[6] 
B. Ducomet and A. Zlotnik, On stability of the CrankNicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. II, Comm. Math. Sci., 5 (2007), 267298. 
[7] 
B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finitedifference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation, Kinetic and Related Models, 2 (2009), 151179. 
[8] 
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6), 4 (2001), 57108. 
[9] 
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987; Abridged English version: "Mathematical Problems and Methods in Hydrodynamical Weather Forecasting," Gordon and Breach, Amsterdam, 2000. 
[10] 
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. 
[11] 
J. Jin and X. Wu, Analysis of finite element method for onedimensional timedependent Schrödinger equation on unbounded domains, J. Comp. Appl. Math., 220 (2008), 240256. doi: 10.1016/j.cam.2007.08.006. 
[12] 
C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension, Am. J. Phys., 72 (2004), 351358. doi: 10.1119/1.1619141. 
[13] 
F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödingertype equations, J. Comp. Phys., 134 (1997), 96107. doi: 10.1006/jcph.1997.5675. 
[14] 
M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equationa compact higher order scheme, Kinetic and Related Models, 1 (2008), 101125. 
[15] 
G. Strang and G. Fix, "An Analysis of the Finite Element Method," PrenticeHall Series in Automatic Computation, PrenticeHall, Inc., Englewood Cliffs, NJ, 1973. 
[16] 
I. A. Zlotnik, Computer simulation of the tunnel effect, (in Russian), Moscow Power Engin. Inst. Bulletin, 6 (2010), 1028. 
[17] 
I. A. Zlotnik, A family of difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a halfstrip, Comput. Math. Math. Phys., 51 (2011), 355376. doi: 10.1134/S0965542511030122. 
show all references
References:
[1] 
X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comp. Phys., 4 (2008), 729796. 
[2] 
X. Antoine and C. Besse, Unconditionally stable discretization schemes of nonreflecting boundary conditions for the onedimensional Schrödinger equation, J. Comp. Phys., 188 (2003), 157175. doi: 10.1016/S00219991(03)001591. 
[3] 
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), 313319. 
[4] 
A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: Fast calculations, approximation, and stability, Comm. Math. Sci., 1 (2003), 501556. 
[5] 
B. Ducomet and A. Zlotnik, On stability of the CrankNicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Comm. Math. Sci., 4 (2006), 741766. 
[6] 
B. Ducomet and A. Zlotnik, On stability of the CrankNicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. II, Comm. Math. Sci., 5 (2007), 267298. 
[7] 
B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finitedifference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation, Kinetic and Related Models, 2 (2009), 151179. 
[8] 
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6), 4 (2001), 57108. 
[9] 
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987; Abridged English version: "Mathematical Problems and Methods in Hydrodynamical Weather Forecasting," Gordon and Breach, Amsterdam, 2000. 
[10] 
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. 
[11] 
J. Jin and X. Wu, Analysis of finite element method for onedimensional timedependent Schrödinger equation on unbounded domains, J. Comp. Appl. Math., 220 (2008), 240256. doi: 10.1016/j.cam.2007.08.006. 
[12] 
C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension, Am. J. Phys., 72 (2004), 351358. doi: 10.1119/1.1619141. 
[13] 
F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödingertype equations, J. Comp. Phys., 134 (1997), 96107. doi: 10.1006/jcph.1997.5675. 
[14] 
M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equationa compact higher order scheme, Kinetic and Related Models, 1 (2008), 101125. 
[15] 
G. Strang and G. Fix, "An Analysis of the Finite Element Method," PrenticeHall Series in Automatic Computation, PrenticeHall, Inc., Englewood Cliffs, NJ, 1973. 
[16] 
I. A. Zlotnik, Computer simulation of the tunnel effect, (in Russian), Moscow Power Engin. Inst. Bulletin, 6 (2010), 1028. 
[17] 
I. A. Zlotnik, A family of difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a halfstrip, Comput. Math. Math. Phys., 51 (2011), 355376. doi: 10.1134/S0965542511030122. 
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