# American Institute of Mathematical Sciences

September  2012, 5(3): 639-667. doi: 10.3934/krm.2012.5.639

## Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation

 1 Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russian Federation 2 Department of Mathematical Modelling, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russian Federation

Received  March 2012 Revised  May 2012 Published  August 2012

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large $x$. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.
Citation: Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639
##### References:
 [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. [2] X. Antoine and C. Besse, Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comp. Phys., 188 (2003), 157-175. doi: 10.1016/S0021-9991(03)00159-1. [3] A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), 313-319. [4] A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: Fast calculations, approximation, and stability, Comm. Math. Sci., 1 (2003), 501-556. [5] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Comm. Math. Sci., 4 (2006), 741-766. [6] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. II, Comm. Math. Sci., 5 (2007), 267-298. [7] B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation, Kinetic and Related Models, 2 (2009), 151-179. [8] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6), 4 (2001), 57-108. [9] V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987; Abridged English version: "Mathematical Problems and Methods in Hydrodynamical Weather Forecasting," Gordon and Breach, Amsterdam, 2000. [10] R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. [11] J. Jin and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domains, J. Comp. Appl. Math., 220 (2008), 240-256. doi: 10.1016/j.cam.2007.08.006. [12] C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension, Am. J. Phys., 72 (2004), 351-358. doi: 10.1119/1.1619141. [13] F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödinger-type equations, J. Comp. Phys., 134 (1997), 96-107. doi: 10.1006/jcph.1997.5675. [14] M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation-a compact higher order scheme, Kinetic and Related Models, 1 (2008), 101-125. [15] G. Strang and G. Fix, "An Analysis of the Finite Element Method," Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. [16] I. A. Zlotnik, Computer simulation of the tunnel effect, (in Russian), Moscow Power Engin. Inst. Bulletin, 6 (2010), 10-28. [17] I. A. Zlotnik, A family of difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a half-strip, Comput. Math. Math. Phys., 51 (2011), 355-376. doi: 10.1134/S0965542511030122.

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##### References:
 [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. [2] X. Antoine and C. Besse, Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comp. Phys., 188 (2003), 157-175. doi: 10.1016/S0021-9991(03)00159-1. [3] A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), 313-319. [4] A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: Fast calculations, approximation, and stability, Comm. Math. Sci., 1 (2003), 501-556. [5] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Comm. Math. Sci., 4 (2006), 741-766. [6] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. II, Comm. Math. Sci., 5 (2007), 267-298. [7] B. Ducomet, A. Zlotnik and I. Zlotnik, On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation, Kinetic and Related Models, 2 (2009), 151-179. [8] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma (6), 4 (2001), 57-108. [9] V. A. Gordin, "Mathematical Problems in Hydrodynamical Weather Forecasting. Computational Aspects," (in Russian), "Gidrometeoizdat," Leningrad, 1987; Abridged English version: "Mathematical Problems and Methods in Hydrodynamical Weather Forecasting," Gordon and Breach, Amsterdam, 2000. [10] R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. [11] J. Jin and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domains, J. Comp. Appl. Math., 220 (2008), 240-256. doi: 10.1016/j.cam.2007.08.006. [12] C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension, Am. J. Phys., 72 (2004), 351-358. doi: 10.1119/1.1619141. [13] F. Schmidt and D. Yevick, Discrete transparent boundary conditions for Schrödinger-type equations, J. Comp. Phys., 134 (1997), 96-107. doi: 10.1006/jcph.1997.5675. [14] M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation-a compact higher order scheme, Kinetic and Related Models, 1 (2008), 101-125. [15] G. Strang and G. Fix, "An Analysis of the Finite Element Method," Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973. [16] I. A. Zlotnik, Computer simulation of the tunnel effect, (in Russian), Moscow Power Engin. Inst. Bulletin, 6 (2010), 10-28. [17] I. A. Zlotnik, A family of difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a half-strip, Comput. Math. Math. Phys., 51 (2011), 355-376. doi: 10.1134/S0965542511030122.
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