American Institute of Mathematical Sciences

Advanced
March  2009, 2(1): 151-179. doi: 10.3934/krm.2009.2.151

On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

Bernard Ducomet 1, , Alexander Zlotnik 2, and  Ilya Zlotnik 3,

1. 

DPTA/Service de Physique Nucléaire, CEA, DAM, DIF, F-91297 Arpajon, France
2. 

Department of Applied Mathematics, Russian State Social University, W. Pieck 4, 129226 Moscow, Russian Federation
3. 

Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation

Received  November 2008 Revised  January 2009 Published  January 2009

We consider a 1D Schrödinger equation with variable coefficients on the half-axis. We study a family of two-level symmetric finite-difference schemes with a three-point parameter dependent averaging in space. This family includes a number of particular schemes. The schemes are coupled to an approximate transparent boundary condition (TBC). We prove two stability bounds with respect to initial data and a free term in the main equation, under suitable conditions on an operator of the approximate TBC. We also consider the family of schemes on an infinite mesh in space. We derive and analyze the discrete TBC allowing to restrict these schemes to a finite mesh and prove the stability conditions for it. Numerical examples are also included.
Keywords: finite-difference schemes, the time-dependent Schrödinger equation, finite element method, approximate and discrete transparent boundary conditions, Stability, discrete convolution operator..
Mathematics Subject Classification: Primary: 65M06, 65M12, 65M60; Secondary: 35Q4.
Citation: Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic & Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151
[1]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
[2]

Lih-Ing W. Roeger. Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 415-429. doi: 10.3934/dcdsb.2008.9.415
[3]

Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems & Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039
[4]

Maike Schulte, Anton Arnold. Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme. Kinetic & Related Models, 2008, 1 (1) : 101-125. doi: 10.3934/krm.2008.1.101
[5]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020222
[6]

Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. Journal of Computational Dynamics, 2020, 7 (2) : 291-312. doi: 10.3934/jcd.2020012
[7]

Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89
[8]

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) : 665-684. doi: 10.3934/dcdsb.2010.13.665
[9]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
[10]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[11]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020234
[12]

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) : 971-987. doi: 10.3934/dcdss.2012.5.971
[13]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
[14]

Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286
[15]

Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050
[16]

Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015
[17]

Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195
[18]

Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems & Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
[19]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[20]

Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences