
Previous Article
Large time behavior of solutions to the nonisentropic compressible NavierStokesPoisson system in $\mathbb{R}^{3}$
 KRM Home
 This Issue

Next Article
Erratum to: Ghost effect by curvature in planar Couette flow [1]
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. 
[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. 
[3] 
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations,, VLSI Design, 6 (1998), 313. 
[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. 
[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. 
[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. 
[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. 
[8] 
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation,, Riv. Mat. Univ. Parma (6), 4 (2001), 57. 
[9] 
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987;, Abridged English version:, (2000). 
[10] 
R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (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), 240. 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), 351. doi: 10.1119/1.1619141. 
[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. 
[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. 
[15] 
G. Strang and G. Fix, "An Analysis of the Finite Element Method,", PrenticeHall Series in Automatic Computation, (1973). 
[16] 
I. A. Zlotnik, Computer simulation of the tunnel effect,, (in Russian), 6 (2010), 10. 
[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. 
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. 
[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. 
[3] 
A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations,, VLSI Design, 6 (1998), 313. 
[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. 
[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. 
[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. 
[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. 
[8] 
M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation,, Riv. Mat. Univ. Parma (6), 4 (2001), 57. 
[9] 
V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987;, Abridged English version:, (2000). 
[10] 
R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (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), 240. 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), 351. doi: 10.1119/1.1619141. 
[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. 
[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. 
[15] 
G. Strang and G. Fix, "An Analysis of the Finite Element Method,", PrenticeHall Series in Automatic Computation, (1973). 
[16] 
I. A. Zlotnik, Computer simulation of the tunnel effect,, (in Russian), 6 (2010), 10. 
[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. 
[1] 
Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finitedifference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151179. doi: 10.3934/krm.2009.2.151 
[2] 
Maike Schulte, Anton Arnold. Discrete transparent boundary conditions for the Schrodinger equation  a compact higher order scheme. Kinetic & Related Models, 2008, 1 (1) : 101125. doi: 10.3934/krm.2008.1.101 
[3] 
Daniele Boffi, Lucia Gastaldi. Discrete models for fluidstructure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems  S, 2016, 9 (1) : 89107. doi: 10.3934/dcdss.2016.9.89 
[4] 
Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems  B, 2010, 13 (3) : 665684. doi: 10.3934/dcdsb.2010.13.665 
[5] 
Holger Teismann. The Schrödinger equation with singular timedependent potentials. Discrete & Continuous Dynamical Systems  A, 2000, 6 (3) : 705722. doi: 10.3934/dcds.2000.6.705 
[6] 
Runchang Lin, Huiqing Zhu. A discontinuous Galerkin leastsquares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489497. doi: 10.3934/proc.2013.2013.489 
[7] 
Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A spacetime discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems  B, 2017, 22 (11) : 128. doi: 10.3934/dcdsb.2017216 
[8] 
Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems  S, 2012, 5 (5) : 971987. doi: 10.3934/dcdss.2012.5.971 
[9] 
Hristo Genev, George Venkov. Soliton and blowup solutions to the timedependent SchrödingerHartree equation. Discrete & Continuous Dynamical Systems  S, 2012, 5 (5) : 903923. doi: 10.3934/dcdss.2012.5.903 
[10] 
Z.G. Feng, K.L. Teo, Y. Zhao. Branch and bound method for sensor scheduling in discrete time. Journal of Industrial & Management Optimization, 2005, 1 (4) : 499512. doi: 10.3934/jimo.2005.1.499 
[11] 
Alexander Zlotnik. The NumerovCrankNicolson scheme on a nonuniform mesh for the timedependent Schrödinger equation on the halfaxis. Kinetic & Related Models, 2015, 8 (3) : 587613. doi: 10.3934/krm.2015.8.587 
[12] 
Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 128. doi: 10.3934/ipi.2018001 
[13] 
Yingwen Guo, Yinnian He. Fully discrete finite element method based on secondorder CrankNicolson/AdamsBashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems  B, 2015, 20 (8) : 25832609. doi: 10.3934/dcdsb.2015.20.2583 
[14] 
Mahboub Baccouch. Superconvergence of the semidiscrete local discontinuous Galerkin method for nonlinear KdVtype problems. Discrete & Continuous Dynamical Systems  B, 2017, 22 (11) : 136. doi: 10.3934/dcdsb.2018104 
[15] 
P. Cerejeiras, U. Kähler, M. M. Rodrigues, N. Vieira. Hodge type decomposition in variable exponent spaces for the timedependent operators: the Schrödinger case. Communications on Pure & Applied Analysis, 2014, 13 (6) : 22532272. doi: 10.3934/cpaa.2014.13.2253 
[16] 
InJee Jeong, Benoit Pausader. Discrete Schrödinger equation and illposedness for the Euler equation. Discrete & Continuous Dynamical Systems  A, 2017, 37 (1) : 281293. doi: 10.3934/dcds.2017012 
[17] 
Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volumefinite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429448. doi: 10.3934/cpaa.2018024 
[18] 
Martin Kružík, Johannes Zimmer. Rateindependent processes with linear growth energies and timedependent boundary conditions. Discrete & Continuous Dynamical Systems  S, 2012, 5 (3) : 591604. doi: 10.3934/dcdss.2012.5.591 
[19] 
E. Fossas, J. M. Olm. Galerkin method and approximate tracking in a nonminimum phase bilinear system. Discrete & Continuous Dynamical Systems  B, 2007, 7 (1) : 5376. doi: 10.3934/dcdsb.2007.7.53 
[20] 
Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339353. doi: 10.3934/mbe.2007.4.339 
2016 Impact Factor: 1.261
Tools
Metrics
Other articles
by authors
[Back to Top]