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Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. (English) Zbl 1053.65072

The paper is devoted to the numerical computation of the solution to the two-dimensional Schrödinger equation in a bounded domain \(\Omega\) with artificial conditions set on the arbitrarily shaped boundary of \(\Omega\) . Using the results of their earlier paper [cf. X. Antoine and C. Besse, J. Math. Pures Appl. 80, No. 7, 701–738 (2001; Zbl 1129.35324)], which allows one to construct a hierarchy of artificial boundary conditions, they study the truncated initial boundary value problem with one of the previous artificial boundary conditions and prove the uniqueness of the solution as well as the decay of the total energy associated with the system.
The wellposedness of the semi-discrete problem discretized by the Crank-Nicolson scheme is investigated. The result indicates that the decay of the energy is preserved at the semi-discrete level, implying hence the stability of the whole scheme. A few numerical experiments are done.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q40 PDEs in connection with quantum mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 1129.35324
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References:

[1] Isaías Alonso-Mallo and Nuria Reguera, Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schrödinger-type equations, SIAM J. Numer. Anal. 40 (2002), no. 1, 134 – 158. · Zbl 1018.65118
[2] Xavier Antoine, Fast approximate computation of a time-harmonic scattered field using the on-surface radiation condition method, IMA J. Appl. Math. 66 (2001), no. 1, 83 – 110. · Zbl 1001.78008
[3] Xavier Antoine and Christophe Besse, Construction, structure and asymptotic approximations of a microdifferential transparent boundary condition for the linear Schrödinger equation, J. Math. Pures Appl. (9) 80 (2001), no. 7, 701 – 738. · Zbl 1129.35324
[4] X. Antoine, and C. Besse, Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comput. Phys. 181 (1) (2003), pp. 157-175. · Zbl 1037.65097
[5] A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design 6 (1-4) (1998), pp. 313-319.
[6] A. Arnold, Mathematical concepts of open quantum boundary conditions, Transport Theory Statist. Phys., 30 (4-6) (2001), pp. 561-584. · Zbl 1019.81010
[7] Anton Arnold and Matthias Ehrhardt, Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics, J. Comput. Phys. 145 (1998), no. 2, 611 – 638. · Zbl 0915.76081
[8] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6) 4* (2001), 57 – 108. Fluid dynamic processes with inelastic interactions at the molecular scale (Torino, 2000). · Zbl 0993.65097
[9] V. A. Baskakov and A. V. Popov, Implementation of transparent boundaries for numerical solution of the Schrödinger equation, Wave Motion 14 (1991), no. 2, 123 – 128. · Zbl 1159.78349
[10] C.-H. Bruneau, L. Di Menza, and T. Lehner, Numerical resolution of some nonlinear Schrödinger-like equations in plasmas, Numer. Methods Partial Differential Equations 15 (1999), no. 6, 672 – 696. , https://doi.org/10.1002/(SICI)1098-2426(199911)15:63.3.CO;2-A · Zbl 0957.76050
[11] F. Schmidt and P. Deuflhard, Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation, Comput. Math. Appl. 29 (1995), no. 9, 53 – 76. · Zbl 0821.65078
[12] Laurent Di Menza, Transparent and absorbing boundary conditions for the Schrödinger equation in a bounded domain, Numer. Funct. Anal. Optim. 18 (1997), no. 7-8, 759 – 775. · Zbl 0895.65041
[13] Thomas Fevens and Hong Jiang, Absorbing boundary conditions for the Schrödinger equation, SIAM J. Sci. Comput. 21 (1999), no. 1, 255 – 282. · Zbl 0938.35013
[14] David Yevick, Tilmann Friese, and Frank Schmidt, A comparison of transparent boundary conditions for the Fresnel equation, J. Comput. Phys. 168 (2001), no. 2, 433 – 444. · Zbl 0990.65099
[15] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 223 – 276. · Zbl 0916.34011
[16] T. Ha-Duong and P. Joly, A principle of images for absorbing boundary conditions, Numer. Methods Partial Differential Equations 10 (1994), no. 4, 411 – 434. · Zbl 0822.65061
[17] Thomas Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta numerica, 1999, Acta Numer., vol. 8, Cambridge Univ. Press, Cambridge, 1999, pp. 47 – 106. · Zbl 0940.65108
[18] Peter Henrici, Fast Fourier methods in computational complex analysis, SIAM Rev. 21 (1979), no. 4, 481 – 527. · Zbl 0416.65022
[19] Y.V. Kopylov, A.V. Popov, and A.V. Vinogradov, Application of the parabolic wave equation to X-ray diffraction optics, Optics Comm. 118 (1995), pp. 619-636.
[20] J.-P. Kuska, Absorbing boundary conditions for the Schrödinger equation on finite intervals, Phys. Rev. B 46 (8) (1992), pp.5000-5003.
[21] Mireille Levy, Parabolic equation methods for electromagnetic wave propagation, IEE Electromagnetic Waves Series, vol. 45, Institution of Electrical Engineers (IEE), London, 2000. · Zbl 0943.78001
[22] Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), no. 3, 704 – 719. · Zbl 0624.65015
[23] Christian Lubich and Achim Schädle, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Comput. 24 (2002), no. 1, 161 – 182. · Zbl 1013.65113
[24] B. Mayfield, Non Local Boundary Conditions for the Schrödinger Equation, Ph.D. Thesis, University of Rhodes Island, Providence, RI, 1989.
[25] Achim Schädle, Non-reflecting boundary conditions for the two-dimensional Schrödinger equation, Wave Motion 35 (2002), no. 2, 181 – 188. · Zbl 1163.74435
[26] Frank Schmidt, Construction of discrete transparent boundary conditions for Schrödinger-type equations, Surveys Math. Indust. 9 (1999), no. 2, 87 – 100. · Zbl 0940.65097
[27] Frank Schmidt and David Yevick, Discrete transparent boundary conditions for Schrödinger-type equations, J. Comput. Phys. 134 (1997), no. 1, 96 – 107. · Zbl 0878.65111
[28] Fred D. Tappert, The parabolic approximation method, Wave propagation and underwater acoustics (Workshop, Mystic, Conn., 1974), Springer, Berlin, 1977, pp. 224 – 287. Lecture Notes in Phys., Vol. 70.
[29] Ahmed I. Zayed, Handbook of function and generalized function transformations, Mathematical Sciences Reference Series, CRC Press, Boca Raton, FL, 1996. · Zbl 0851.44002
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