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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), 729. Google Scholar 
[2] 
X. Antoine and C. Besse, Unconditionally stable discretization schemes of nonreflecting boundary conditions for the onedimensional Schrödinger equation,, J. Comp. Phys., 188 (2003), 157. doi: 10.1016/S00219991(03)001591. Google Scholar 
[3] 
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations,, VLSI Design, 6 (1998), 313. Google Scholar 
[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), 501. Google Scholar 
[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), 741. Google Scholar 
[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), 267. Google Scholar 
[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), 151. Google Scholar 
[8] 
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation,, Riv. Mat. Univ. Parma (6), 4 (2001), 57. Google Scholar 
[9] 
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987;, Abridged English version:, (2000). Google Scholar 
[10] 
R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (1985). Google Scholar 
[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), 240. doi: 10.1016/j.cam.2007.08.006. Google Scholar 
[12] 
C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension,, Am. J. Phys., 72 (2004), 351. doi: 10.1119/1.1619141. Google Scholar 
[13] 
F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödingertype equations,, J. Comp. Phys., 134 (1997), 96. doi: 10.1006/jcph.1997.5675. Google Scholar 
[14] 
M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equationa compact higher order scheme,, Kinetic and Related Models, 1 (2008), 101. Google Scholar 
[15] 
G. Strang and G. Fix, "An Analysis of the Finite Element Method,", PrenticeHall Series in Automatic Computation, (1973). Google Scholar 
[16] 
I. A. Zlotnik, Computer simulation of the tunnel effect,, (in Russian), 6 (2010), 10. Google Scholar 
[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), 355. doi: 10.1134/S0965542511030122. Google Scholar 
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), 729. Google Scholar 
[2] 
X. Antoine and C. Besse, Unconditionally stable discretization schemes of nonreflecting boundary conditions for the onedimensional Schrödinger equation,, J. Comp. Phys., 188 (2003), 157. doi: 10.1016/S00219991(03)001591. Google Scholar 
[3] 
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations,, VLSI Design, 6 (1998), 313. Google Scholar 
[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), 501. Google Scholar 
[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), 741. Google Scholar 
[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), 267. Google Scholar 
[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), 151. Google Scholar 
[8] 
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation,, Riv. Mat. Univ. Parma (6), 4 (2001), 57. Google Scholar 
[9] 
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987;, Abridged English version:, (2000). Google Scholar 
[10] 
R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (1985). Google Scholar 
[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), 240. doi: 10.1016/j.cam.2007.08.006. Google Scholar 
[12] 
C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension,, Am. J. Phys., 72 (2004), 351. doi: 10.1119/1.1619141. Google Scholar 
[13] 
F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödingertype equations,, J. Comp. Phys., 134 (1997), 96. doi: 10.1006/jcph.1997.5675. Google Scholar 
[14] 
M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equationa compact higher order scheme,, Kinetic and Related Models, 1 (2008), 101. Google Scholar 
[15] 
G. Strang and G. Fix, "An Analysis of the Finite Element Method,", PrenticeHall Series in Automatic Computation, (1973). Google Scholar 
[16] 
I. A. Zlotnik, Computer simulation of the tunnel effect,, (in Russian), 6 (2010), 10. Google Scholar 
[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), 355. doi: 10.1134/S0965542511030122. Google Scholar 
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