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Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip

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Abstract

An initial-boundary value problem for the generalized Schrödinger equation in a semi-infinite strip is solved. A new family of two-level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed, and numerical results are presented.

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REFERENCES

  1. J.-F. Berger, M. Girod, and D. Gogny, “Time-Dependent Quantum Collective Dynamics Applied to Nuclear Fission,” Comp. Phys. Commun. 63, 365–374 (1991).

    Article  MATH  Google Scholar 

  2. H. Goutte, J.-F. Berger, P. Casoly, and D. Gogny, “Microscopic Approach of Fission Dynamics Applied to Fragment Kinetic Energy and Mass Distribution in 238U,” Phys. Rev. C 71((1–13)), 4316 (2005).

    Google Scholar 

  3. 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, 729–796 (2008).

    MathSciNet  Google Scholar 

  4. A. Arnold, “Numerically Absorbing Boundary Conditions for Quantum Evolution Equations,” VLSI Design 6, 313–319 (1998).

    Article  Google Scholar 

  5. M. Ehrhardt and A. Arnold, “Discrete Transparent Boundary Conditions for the Schrödinger Equation,” Riv. Mat. Univ. Parma 6, 57–108 (2001).

    MathSciNet  Google Scholar 

  6. A. Arnold, M. Ehrhardt, and I. Sofronov, “Discrete Transparent Boundary Conditions for the Schrödinger Equation: Fast Calculations, Approximation and Stability,” Commun. Math. Sci. 1, 501–556 (2003).

    MATH  MathSciNet  Google Scholar 

  7. B. Ducomet and A. Zlotnik, “On Stability of the Crank-Nicolson Scheme with Approximate Transparent Boundary Conditions for the Schrödinger Equation. Part I,” Commun. Math. Sci. 4, 741–766 (2006).

    MATH  MathSciNet  Google Scholar 

  8. B. Ducomet and A. Zlotnik, “On Stability of the Crank-Nicolson Scheme with Approximate Transparent Boundary Conditions for the Schrödinger Equation. Part II,” Commun. Math. Sci. 5, 267–298 (2007).

    MATH  MathSciNet  Google Scholar 

  9. A. A. Zlotnik and B. Ducomet, “Stability of a Symmetric Finite-Difference Scheme with Approximate Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation,” Dokl. Akad. Nauk 413, 96–107 (2007) [Dokl. Math. 75, 260–265 (2007)].

    MathSciNet  Google Scholar 

  10. F. Schmidt and D. Yevick, “Discrete Transparent Boundary Conditions for Schrödinger-Type Equations,” J. Comput. Phys. 134, 96–107 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  11. X. Antoine and C. Besse, “Unconditionally Stable Discretization Schemes of Non-Reflecting Boundary Conditions for the One-Dimensional Schrödinger Equation,” J. Comput. Phys. 188, 157–175 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Han, J. Jin, and X. Wu, “A Finite-Difference Method for the One-Dimensional Time-Dependent Schrödinger-Type Equation on Unbounded Domains,” Comput. Math. Appl. 50, 1345–1362 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  13. C. A. Moyer, “Numerov Extension of Transparent Boundary Conditions for the Schrödinger Equation Discretized in One Dimension,” Am. J. Phys. 72, 351–358 (2004).

    Article  Google Scholar 

  14. M. Schulte and A. Arnold, “Discrete Transparent Boundary Conditions for the Schrödinger Equation, a Compact Higher Order Scheme,” Kinetic Related Models 1(1), 101–125 (2008).

    MATH  MathSciNet  Google Scholar 

  15. A. A. Zlotnik and I. A. Zlotnik, “On the Stability of a Family of Finite-Difference Schemes with Approximate Transparent Boundary Conditions for the Schrödinger Equations in a Semi-Infinite Strip,” Vestn. Mosk. Energ. Inst., No. 6, 31–45 (2008).

  16. B. Ducomet, A. Zlotnik, and I. Zlotnik, “On a Family of Finite-Difference Schemes with Discrete Transparent Boundary Conditions for a Generalized Schrödinger Equation,” Kinetic Related Models 2(1), 151–180 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  17. I. A. Zlotnik, “On the Stability of a Family of Finite-Difference Schemes with Approximate Transparent Boundary Conditions for the Nonstationary Schrödinger Equations in a Semi-Infinite Strip,” Vestn. Mosk. Energ. Inst., No. 6, 127–144 (2009).

  18. A. A. Zlotnik, “Some Finite-Element and Finite-Difference Methods for Solving Mathematical Physics Problems with Nonsmooth Data in N-Dimensional Cube,” Sov. J. Numer. Anal. Math. Model. 6, 421–451 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  19. I. A. Zlotnik, “Computer Simulation of the Tunnel Effect,” Vestn. Mosk. Energ. Inst., No. 6, 10–28 (2010).

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Authors and Affiliations

  1. Department of Mathematical Modeling, Moscow Power Engineering Institute (Technical University), Krasnokazarmennaya ul. 14, Moscow, 111250, Russia

    I. A. Zlotnik

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  1. I. A. Zlotnik
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Correspondence to I. A. Zlotnik.

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Original Russian Text © I.A. Zlotnik, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 3, pp. 384–406.

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Zlotnik, I.A. Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip. Comput. Math. and Math. Phys. 51, 355–376 (2011). https://doi.org/10.1134/S0965542511030122

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