# The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

Bernard Ducomet; Alexander Zlotnik; Ilya Zlotnik

- Volume: 48, Issue: 6, page 1681-1699
- ISSN: 0764-583X

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topDucomet, Bernard, Zlotnik, Alexander, and Zlotnik, Ilya. "The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.6 (2014): 1681-1699. <http://eudml.org/doc/273296>.

@article{Ducomet2014,

abstract = {We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank–Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time L2-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method.},

author = {Ducomet, Bernard, Zlotnik, Alexander, Zlotnik, Ilya},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {the time-dependent Schrödinger equation; the Crank–Nicolson finite-difference scheme; the strang splitting; approximate and discrete transparent boundary conditions; stability; tunnel effect; time-dependent Schrödinger equation; Crank-Nicolson finite-difference scheme; strang splitting; initial boundary value problem; algorithm; fast Fourier transform; numerical result; error analysis},

language = {eng},

number = {6},

pages = {1681-1699},

publisher = {EDP-Sciences},

title = {The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip},

url = {http://eudml.org/doc/273296},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Ducomet, Bernard

AU - Zlotnik, Alexander

AU - Zlotnik, Ilya

TI - The splitting in potential Crank–Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 6

SP - 1681

EP - 1699

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

LA - eng

KW - the time-dependent Schrödinger equation; the Crank–Nicolson finite-difference scheme; the strang splitting; approximate and discrete transparent boundary conditions; stability; tunnel effect; time-dependent Schrödinger equation; Crank-Nicolson finite-difference scheme; strang splitting; initial boundary value problem; algorithm; fast Fourier transform; numerical result; error analysis

UR - http://eudml.org/doc/273296

ER -

## References

top- [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) 729–796. MR2463189
- [2] X. Antoine, C. Besse and V. Mouysset, Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Math. Comp.73 (2004) 1779–1799. Zbl1053.65072MR2059736
- [3] A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: fast calculations, approximation and stability. Commun. Math. Sci.1 (2003) 501–556. Zbl1085.65513MR2069942
- [4] J.F. Berger, M. Girod and D. Gogny, Microscopic analysis of collective dynamics in low energy fission. Nuclear Physics A428 (1984) 23–36.
- [5] J.-F. Berger, M. Girod and D. Gogny, Time-dependent quantum collective dynamics applied to nuclear fission. Comp. Phys. Commun.63 (1991) 365–374. Zbl0850.65332
- [6] S. Blanes and P.C. Moan, Splitting methods for the time-dependent Schrödinger equation. Phys. Lett. A265 (2000) 35–42. Zbl0941.81025MR1739100
- [7] C.R. Chinn, J.F. Berger, D. Gogny and M.S. Weiss, Limits on the lifetime of the shape isomer of 238U. Phys. Rev. C45 (1984) 1700–1708.
- [8] L. Di Menza, Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain. Numer. Funct. Anal. Optimiz.18 (1997) 759–775. Zbl0895.65041MR1472153
- [9] 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 (2006) 741–766. Zbl1119.65085MR2264818
- [10] 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 (2007) 267–298. Zbl1141.65069MR2334843
- [11] 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). MR3316597
- [12] B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finite-difference schemes with discrete transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic Relat. Models2 (2009), 151–179. Zbl1195.65106MR2472153
- [13] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma6 (2001) 57–108. Zbl0993.65097MR1881105
- [14] Z. Gao and S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. Appl. Numer. Math.61 (2011) 593–614. Zbl1221.65220MR2754579
- [15] L. Gauckler, Convergence of a split-step Hermite method for Gross-Pitaevskii equation. IMA J. Numer. Anal.31 (2011) 396–415. Zbl1223.65079MR2813177
- [16] 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 (2005) 4316.
- [17] H. Hofmann, Quantummechanical treatment of the penetration through a two-dimensional fission barrier. Nuclear Physics A224 (1974) 116–139.
- [18] C. Lubich, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput.77 (2008) 2141–2153. Zbl1198.65186MR2429878
- [19] C. Lubich, From quantum to classical molecular dynamics. Reduced models and numerical analysis. Zürich Lect. Adv. Math. EMS, Zürich (2008). Zbl1160.81001MR2474331
- [20] C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numer. Math.49 (2009) 199–215. Zbl1162.65385MR2486135
- [21] P. Ring, H. Hassman and J.O. Rasmussen, On the treatment of a two-dimensional fission model with complex trajectories. Nuclear Physics A296 (1978) 50–76.
- [22] P. Ring and P. Schuck, The nuclear many-body problem. Theoret. Math. Phys. Springer-Verlag, New York, Heidelberg, Berlin (1980). MR611683
- [23] S.G. Rohozinski, J. Dobaczewski, B. Nerlo-Pomorska, K. Pomorski and J. Srebny, Microscopic dynamic calculations of collective states in Xenon and Barium isotopes. Nuclear Physics A292 (1978) 66–87.
- [24] A. Schädle, Non-reflecting boundary conditions for the two-dimensional Schrödinger equation. Wave Motion35 (2002) 181–188. Zbl1163.74435MR1869342
- [25] M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, a compact higher order scheme. Kinetic Relat. Models1 (2008) 101–125. Zbl1145.65058MR2383718
- [26] G. Strang, On the construction and comparison of difference scheme. SIAM J. Numer. Anal.5 (1968) 506–517. Zbl0184.38503MR235754
- [27] J. Szeftel, Design of absorbing boundary conditions for Schrödinger equations in Rd. SIAM J. Numer. Anal.42 (2004) 1527–1551. Zbl1094.35037MR2114289
- [28] N.N. Yanenko, The method of fractional steps: solution of problems of mathematical physics in several variables. Springer, New York (1971). Zbl0209.47103MR307493
- [29] S.B. Zaitseva and A.A. Zlotnik, Error analysis in L2(Q) for symmetrized locally one-dimensional methods for the heat equation. Russ. J. Numer. Anal. Math. Model.13 (1998) 69–91. Zbl0943.65104MR1611638
- [30] S.B. Zaitseva and A.A. Zlotnik, Sharp error analysis of vector splitting methods for the heat equation. Comput. Math. Phys.39 (1999) 448–467. Zbl0970.65099MR1683579
- [31] A.A. Zlotnik, 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. Zbl0816.65063MR1137008
- [32] A. Zlotnik and S. Ilyicheva, Sharp error bounds for a symmetrized locally 1D method for solving the 2D heat equation. Comput. Meth. Appl. Math.6 (2006) 94–114. Zbl1094.65094MR2220487
- [33] 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). Zbl1326.65112MR3323462
- [34] A.A. Zlotnik and I.A. Zlotnik, 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. Zbl1236.65116MR2810155
- [35] I.A. Zlotnik, Computer simulation of the tunnel effect. Moscow Power Engin. Inst. Bulletin 17 (2010) 10–28 (in Russian).
- [36] 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. Phys.51 (2011) 355–376. Zbl1249.35313MR2839572

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