Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T21:00:57.019Z Has data issue: false hasContentIssue false
  • English
  • Français

The splitting in potential Crank–Nicolson scheme with discretetransparent boundary conditions for the Schrödinger equation on a semi-infinitestrip

Published online by Cambridge University Press:  24 September 2014

Bernard Ducomet ,
Alexander Zlotnik  and
Ilya Zlotnik
Bernard Ducomet
Affiliation:
CEA, DAM, DIF, 91297, Arpajon, France. . bernard.ducomet@cea.fr
Alexander Zlotnik
Affiliation:
Department of Higher Mathematics at Faculty of Economics, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russia.; azlotnik2008@gmail.com
Ilya Zlotnik
Affiliation:
Department of Mathematical Modelling, National Research University Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russia. ; ilya.zlotnik@gmail.com
Get access

Abstract

We consider an initial-boundary value problem for a generalized 2D time-dependentSchrödinger equation (with variable coefficients) on a semi-infinite strip. For theCrank–Nicolson-type finite-difference scheme with approximate or discrete transparentboundary conditions (TBCs), the Strang-type splitting with respect to the potential isapplied. For the resulting method, the unconditional uniform in time L2-stability isproved. Due to the splitting, an effective direct algorithm using FFT is developed now toimplement the method with the discrete TBC for general potential. Numerical results on thetunnel effect for rectangular barriers are included together with the detailed practicalerror analysis confirming nice properties of the method.

Keywords

The time-dependent Schrödinger equationthe Crank–Nicolson finite-difference schemethe Strang splittingapproximate and discrete transparent boundary conditionsstabilitytunnel effect
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1681 - 1699
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antoine, X., Arnold, A., Besse, C., Ehrhardt, M. and Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comp. Phys. 4 (2008) 729–796. Google Scholar
Antoine, X., Besse, C. and Mouysset, V., Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Math. Comp. 73 (2004) 1779–1799. Google Scholar
Arnold, A., Ehrhardt, M. and Sofronov, I., Discrete transparent boundary conditions for the Schrödinger equation: fast calculations, approximation and stability. Commun. Math. Sci. 1 (2003) 501–556. Google Scholar
Berger, J.F., Girod, M. and Gogny, D., Microscopic analysis of collective dynamics in low energy fission. Nuclear Physics A 428 (1984) 23–36. Google Scholar
Berger, J.-F., Girod, M. and Gogny, D., Time-dependent quantum collective dynamics applied to nuclear fission. Comp. Phys. Commun. 63 (1991) 365–374. Google Scholar
Blanes, S. and Moan, P.C., Splitting methods for the time-dependent Schrödinger equation. Phys. Lett. A 265 (2000) 35–42. Google Scholar
Chinn, C.R., Berger, J.F., Gogny, D. and Weiss, M.S., Limits on the lifetime of the shape isomer of 238U. Phys. Rev. C 45 (1984) 1700–1708. Google Scholar
Di Menza, L., Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain. Numer. Funct. Anal. Optimiz. 18 (1997) 759–775. Google Scholar
Ducomet, B. and Zlotnik, A., On stability of the Crank–Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part I. Commun. Math. Sci. 4 (2006) 741–766. Google Scholar
Ducomet, B. and Zlotnik, A., On stability of the Crank–Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part II. Commun. Math. Sci. 5 (2007) 267–298. Google Scholar
B. Ducomet, A. Zlotnik and A. Romanova, On a splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped. Appl. Math. Comp. To appear (2014).
Ducomet, B., Zlotnik, A. and Zlotnik, I., On a family of finite-difference schemes with discrete transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic Relat. Models 2 (2009), 151–179. Google Scholar
Ehrhardt, M. and Arnold, A., Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma 6 (2001) 57–108. Google Scholar
Gao, Z. and Xie, S., Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. Appl. Numer. Math. 61 (2011) 593–614. Google Scholar
Gauckler, L., Convergence of a split-step Hermite method for Gross-Pitaevskii equation. IMA J. Numer. Anal. 31 (2011) 396–415. Google Scholar
Goutte, H., Berger, J.-F., Casoly, P. and Gogny, D., Microscopic approach of fission dynamics applied to fragment kinetic energy and mass distribution in 238U. Phys. Rev. C 71 (2005) 4316. Google Scholar
Hofmann, H., Quantummechanical treatment of the penetration through a two-dimensional fission barrier. Nuclear Physics A 224 (1974) 116–139. Google Scholar
Lubich, C., On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008) 2141–2153. Google Scholar
C. Lubich, From quantum to classical molecular dynamics. Reduced models and numerical analysis. Zürich Lect. Adv. Math. EMS, Zürich (2008).
Neuhauser, C. and Thalhammer, M., On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numer. Math. 49 (2009) 199–215. Google Scholar
Ring, P., Hassman, H. and Rasmussen, J.O., On the treatment of a two-dimensional fission model with complex trajectories. Nuclear Physics A 296 (1978) 50–76. Google Scholar
P. Ring and P. Schuck, The nuclear many-body problem. Theoret. Math. Phys. Springer-Verlag, New York, Heidelberg, Berlin (1980).
Rohozinski, S.G., Dobaczewski, J., Nerlo-Pomorska, B., Pomorski, K. and Srebny, J., Microscopic dynamic calculations of collective states in Xenon and Barium isotopes. Nuclear Physics A 292 (1978) 66–87. Google Scholar
Schädle, A., Non-reflecting boundary conditions for the two-dimensional Schrödinger equation. Wave Motion 35 (2002) 181–188. Google Scholar
Schulte, M. and Arnold, A., Discrete transparent boundary conditions for the Schrödinger equation, a compact higher order scheme. Kinetic Relat. Models 1 (2008) 101–125. Google Scholar
Strang, G., On the construction and comparison of difference scheme. SIAM J. Numer. Anal. 5 (1968) 506–517. Google Scholar
Szeftel, J., Design of absorbing boundary conditions for Schrödinger equations in Rd. SIAM J. Numer. Anal. 42 (2004) 1527–1551. Google Scholar
N.N. Yanenko, The method of fractional steps: solution of problems of mathematical physics in several variables. Springer, New York (1971).
Zaitseva, S.B. and Zlotnik, A.A., Error analysis in L 2(Q) for symmetrized locally one-dimensional methods for the heat equation. Russ. J. Numer. Anal. Math. Model. 13 (1998) 69–91. Google Scholar
Zaitseva, S.B. and Zlotnik, A.A., Sharp error analysis of vector splitting methods for the heat equation. Comput. Math. Phys. 39 (1999) 448–467. Google Scholar
Zlotnik, A.A., Some finite-element and finite-difference methods for solving mathematical physics problems with non-smooth data in n-dimensional cube. Sov. J. Numer. Anal. Math. Modell. 6 (1991) 421–451. Google Scholar
Zlotnik, A. and Ilyicheva, S., Sharp error bounds for a symmetrized locally 1D method for solving the 2D heat equation. Comput. Meth. Appl. Math. 6 (2006) 94–114. Google Scholar
A. Zlotnik and 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. To appear (2014).
Zlotnik, A.A. and Zlotnik, I.A., Family of finite-difference schemes with transparent boundary conditions for the nonstationary Schrödinger equation in a semi-infinite strip. Dokl. Math. 83 (2011) 12–18. Google Scholar
Zlotnik, I.A., Computer simulation of the tunnel effect. Moscow Power Engin. Inst. Bulletin 17 (2010) 10–28 (in Russian). Google Scholar
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. Phys. 51 (2011) 355–376. Google Scholar