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Splitting in Potential Finite-Difference Schemes with Discrete Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation

  • Alexander ZlotnikEmail author
  • Bernard Ducomet
  • Ilya Zlotnik
  • Alla Romanova
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

The time-dependent Schrödinger equation is the key one in many fields. It should be often solved in unbounded space domains. Several approaches are known to deal with such problems using approximate transparent boundary conditions (TBCs) on the artificial boundaries. Among them, there exist the so-called discrete TBCs whose advantages are the complete absence of spurious reflections, reliable computational stability, clear mathematical background and the corresponding rigorous stability theory. In this paper, the Strang-type splitting with respect to the potential is applied to three two-level schemes with different discretizations in space having the approximation order O(τ 2 + | h |  k ), k = 2 or 4. Explicit forms of the discrete TBCs are given and results on existence, uniqueness and uniform in time L 2-stability of solutions are stated in a unified manner. Due to splitting, an effective direct algorithm to implement the schemes is presented for general potential.

Keywords

Approximation Order Splitting Scheme High Order Scheme Discrete Convolution Transparent Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    X. Antoine, A. Arnold, C. Besse et al., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4(4), 729–796 (2008)MathSciNetGoogle Scholar
  2. 2.
    A. Arnold, M. Ehrhardt, I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: fast calculations, approximation and stability. Commun. Math. Sci. 1(3), 501–556 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    B. Ducomet, A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part I. Commun. Math. Sci. 4(4), 741–766 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    B. Ducomet, A. Zlotnik, A. Romanova, On a splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped. Appl. Math. Comput. (2015, in press)Google Scholar
  5. 5.
    B. Ducomet, A. Zlotnik, I. Zlotnik, On a family of finite-difference schemes with discrete transparent boundary conditions for a generalized 1D Schrödinger equation. Kinet. Relat. Models 2(1), 151–179 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    ___________________________________________________________________ , The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip. ESAIM: Math. Model. Numer. Anal. 48(6), 1681–1699 (2014)Google Scholar
  7. 7.
    M. Ehrhardt, A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma 6, 57–108 (2001)MathSciNetGoogle Scholar
  8. 8.
    C. Lubich, From Quantum to Classical Molecular Dynamics. Reduced Models and Numerical Analysis (EMS, Zürich, 2008)Google Scholar
  9. 9.
    M. Schulte, A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, a compact higher order scheme. Kinet. Relat. Models 1(1), 101–125 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    A.A. Zlotnik, A.V. Lapukhina, Stability of a Numerov type finite-difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis. J. Math. Sci. 169(1), 84–97 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    A. Zlotnik, A. Romanova, On a Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip. Appl. Numer. Math. (2015)Google Scholar
  12. 12.
    I.A. Zlotnik, Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip. Comput. Math. Math. Phys. 51(3), 355–376 (2011)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander Zlotnik
    • 1
    Email author
  • Bernard Ducomet
    • 2
  • Ilya Zlotnik
    • 3
  • Alla Romanova
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.DPTA/Service de Physique NucléaireCEA/DAM/DIF Ile de FranceArpajonFrance
  3. 3.Settlement Depository CompanyMoscowRussia

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