# American Institute of Mathematical Sciences

September  2015, 8(3): 587-613. doi: 10.3934/krm.2015.8.587

## The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis

 1 Department of Mathematics at Faculty of Economics Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow

Received  November 2014 Revised  March 2015 Published  June 2015

We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The finite-difference scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs). Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes. In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh. Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure higher order in time as well.
Citation: Alexander Zlotnik. The Numerov-Crank-Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis. Kinetic & Related Models, 2015, 8 (3) : 587-613. doi: 10.3934/krm.2015.8.587
##### 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. Comput. Phys., 4 (2008), 729. [2] A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations,, VLSI Design, 6 (1998), 313. doi: 10.1155/1998/38298. [3] H. S. Arora and Y. Miyamoto, Portable scheme for solving 1-D time-dependent Schrödinger equation for photo-induced dynamics of an electron in quantum wells,, IEEE J. Quantum Electronics, 49 (2013), 395. [4] S. Chen, X. Gao, J. Li, A. Becker and A. Jaroń-Becker, Application of a numerical-basis-state method to strong-field excitation and ionization of hydrogen atoms,, Phys. Rev. A, 86 (2012). doi: 10.1103/PhysRevA.86.013410. [5] S. A. Chin and J. Geiser, Multi-product operator splitting as a general method of solving autonomous and nonautonomous equations,, IMA J. Numer. Anal., 31 (2011), 1552. doi: 10.1093/imanum/drq022. [6] 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. doi: 10.4310/CMS.2006.v4.n4.a4. [7] 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. doi: 10.4310/CMS.2007.v5.n2.a3. [8] 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. Comput., 255 (2015), 196. doi: 10.1016/j.amc.2014.07.058. [9] 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. Models, 2 (2009), 151. doi: 10.3934/krm.2009.2.151. [10] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation,, Riv. Mat. Univ. Parma, 6 (2001), 57. [11] I. Farago, A. Havasi and Z. Zlatev, Richardson-extrapolated sequential splitting and its application,, J. Comput. Appl. Math., 226 (2009), 218. doi: 10.1016/j.cam.2008.08.003. [12] B. Gustafsson, High Order Difference Methods for Time Dependent PDE,, Springer, (2008). [13] E. Hairer, Ch. Lubich and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations,, SIAM J. Sci. Stat. Comput., 6 (1985), 532. doi: 10.1137/0906037. [14] A. Heidari, O. A. Beg and M. Ghorbani, Study of the vibrational characteristics of the homonuclear diatomic nuclear Schrödinger equation with a Numerov method using a number of empirical potential functions,, Russ. J. Phys. Chem. A, 87 (2013), 216. doi: 10.1134/S0036024413020040. [15] M. K. Jain, S. R. K. Iyengar and G. S. Subramanyam, Variable mesh methods for the numerical solution of two-point singular perturbation problems,, Comput. Meth. Appl. Mech. Engrg., 42 (1984), 273. doi: 10.1016/0045-7825(84)90009-4. [16] J. Jin and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain,, J. Comput. Appl. Math., 220 (2008), 240. doi: 10.1016/j.cam.2007.08.006. [17] C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension,, Amer. J. Phys., 72 (2004), 351. [18] M. Radziunas, R. Čiegis and A. Mirinavičus, On compact higher order finite difference schemes for linear Schrödinger problem on non-uniform meshes,, Int. J. Numer. Anal. Model., 11 (2014), 303. [19] M. Rizea, Exponential fitting method for the time-dependent Schrödinger equation,, J. Math. Chem., 48 (2010), 55. doi: 10.1007/s10910-009-9626-1. [20] F. Robicheaux, Low-energy scattering of molecules and ions in a magnetic field,, Phys. Rev. A, 89 (2014). doi: 10.1103/PhysRevA.89.062701. [21] F. Schmidt and P. Deuflhard, Discrete transparent boundary conditions for the numerical solution of Fresnel's equation,, Comput. Math. Appl., 29 (1995), 53. doi: 10.1016/0898-1221(95)00037-Y. [22] M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, a compact higher order scheme,, Kinetic Relat. Models, 1 (2008), 101. doi: 10.3934/krm.2008.1.101. [23] T. E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation,, J. Math. Chem., 46 (2009), 981. doi: 10.1007/s10910-009-9553-1. [24] K. Singer, U. Poschinger and M. Murphy, et al., Colloquium: Trapped ions as quantum bits: Essential numerical tools,, Rev. Mod. Phys., 82 (2010). doi: 10.1103/RevModPhys.82.2609. [25] B. A. Stickler and E. Schachinger, The one-dimensional stationary Schrödinger equation,, in Basic Concepts in Computational Physics, (2014), 131. doi: 10.1007/978-3-319-02435-6. [26] Z.-Z. Sun, The stability and convergence of an explicit difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions,, J. Comput. Phys., 219 (2006), 879. doi: 10.1016/j.jcp.2006.07.001. [27] B. R. Wong, Numerical solution of the time-dependent Schrödinger equation,, in Frontiers in Physics, (2009), 396. [28] S.-S. Xie, G.-X. Li and S. Yi, Compact finite difference schemes with high accuracy for one-dimensional Schrödinger equation,, Comput. Meth. Appl. Mech. Engrg., 198 (2009), 1052. doi: 10.1016/j.cma.2008.11.011. [29] A. A. Zlotnik, Convergence rate estimates of finite-element methods for second-order hyperbolic equations,, in Numerical Methods and Applications (ed. G.I. Marchuk), (1994), 155. [30] A. A. Zlotnik, Error estimates of the Crank-Nicolson-polylinear FEM with the discrete TBC for the generalized Schrödinger equation in an unbounded parallelepiped,, in Finite Difference Methods, (2014), 18. [31] A. A. Zlotnik and A. V. Lapukhina, Stability of a Numerov type finite-difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis,, J. Math. Sci., 169 (2010), 84. doi: 10.1007/s10958-010-0040-9. [32] 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., 93 (2015), 279. doi: 10.1016/j.apnum.2014.05.003. [33] A. Zlotnik and I. Zlotnik, Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation,, Kinetic Relat. Models, 5 (2012), 639. doi: 10.3934/krm.2012.5.639. [34] A. Zlotnik and I. Zlotnik, The high order method with discrete TBCs for solving the Cauchy problem for the 1D Schrödinger equation,, Comput. Meth. Appl. Math., 15 (2015), 233. doi: 10.1515/cmam-2015-0007. [35] A. Zlotnik and I. Zlotnik, Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis,, Russ. J. Numer. Anal. Math. Model., 31 (2016).

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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. Comput. Phys., 4 (2008), 729. [2] A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations,, VLSI Design, 6 (1998), 313. doi: 10.1155/1998/38298. [3] H. S. Arora and Y. Miyamoto, Portable scheme for solving 1-D time-dependent Schrödinger equation for photo-induced dynamics of an electron in quantum wells,, IEEE J. Quantum Electronics, 49 (2013), 395. [4] S. Chen, X. Gao, J. Li, A. Becker and A. Jaroń-Becker, Application of a numerical-basis-state method to strong-field excitation and ionization of hydrogen atoms,, Phys. Rev. A, 86 (2012). doi: 10.1103/PhysRevA.86.013410. [5] S. A. Chin and J. Geiser, Multi-product operator splitting as a general method of solving autonomous and nonautonomous equations,, IMA J. Numer. Anal., 31 (2011), 1552. doi: 10.1093/imanum/drq022. [6] 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. doi: 10.4310/CMS.2006.v4.n4.a4. [7] 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. doi: 10.4310/CMS.2007.v5.n2.a3. [8] 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. Comput., 255 (2015), 196. doi: 10.1016/j.amc.2014.07.058. [9] 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. Models, 2 (2009), 151. doi: 10.3934/krm.2009.2.151. [10] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation,, Riv. Mat. Univ. Parma, 6 (2001), 57. [11] I. Farago, A. Havasi and Z. Zlatev, Richardson-extrapolated sequential splitting and its application,, J. Comput. Appl. Math., 226 (2009), 218. doi: 10.1016/j.cam.2008.08.003. [12] B. Gustafsson, High Order Difference Methods for Time Dependent PDE,, Springer, (2008). [13] E. Hairer, Ch. Lubich and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations,, SIAM J. Sci. Stat. Comput., 6 (1985), 532. doi: 10.1137/0906037. [14] A. Heidari, O. A. Beg and M. Ghorbani, Study of the vibrational characteristics of the homonuclear diatomic nuclear Schrödinger equation with a Numerov method using a number of empirical potential functions,, Russ. J. Phys. Chem. A, 87 (2013), 216. doi: 10.1134/S0036024413020040. [15] M. K. Jain, S. R. K. Iyengar and G. S. Subramanyam, Variable mesh methods for the numerical solution of two-point singular perturbation problems,, Comput. Meth. Appl. Mech. Engrg., 42 (1984), 273. doi: 10.1016/0045-7825(84)90009-4. [16] J. Jin and X. Wu, Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain,, J. Comput. Appl. Math., 220 (2008), 240. doi: 10.1016/j.cam.2007.08.006. [17] C. A. Moyer, Numerov extension of transparent boundary conditions for the Schrödinger equation discretized in one dimension,, Amer. J. Phys., 72 (2004), 351. [18] M. Radziunas, R. Čiegis and A. Mirinavičus, On compact higher order finite difference schemes for linear Schrödinger problem on non-uniform meshes,, Int. J. Numer. Anal. Model., 11 (2014), 303. [19] M. Rizea, Exponential fitting method for the time-dependent Schrödinger equation,, J. Math. Chem., 48 (2010), 55. doi: 10.1007/s10910-009-9626-1. [20] F. Robicheaux, Low-energy scattering of molecules and ions in a magnetic field,, Phys. Rev. A, 89 (2014). doi: 10.1103/PhysRevA.89.062701. [21] F. Schmidt and P. Deuflhard, Discrete transparent boundary conditions for the numerical solution of Fresnel's equation,, Comput. Math. Appl., 29 (1995), 53. doi: 10.1016/0898-1221(95)00037-Y. [22] M. Schulte and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation, a compact higher order scheme,, Kinetic Relat. Models, 1 (2008), 101. doi: 10.3934/krm.2008.1.101. [23] T. E. Simos, A new Numerov-type method for the numerical solution of the Schrödinger equation,, J. Math. Chem., 46 (2009), 981. doi: 10.1007/s10910-009-9553-1. [24] K. Singer, U. Poschinger and M. Murphy, et al., Colloquium: Trapped ions as quantum bits: Essential numerical tools,, Rev. Mod. Phys., 82 (2010). doi: 10.1103/RevModPhys.82.2609. [25] B. A. Stickler and E. Schachinger, The one-dimensional stationary Schrödinger equation,, in Basic Concepts in Computational Physics, (2014), 131. doi: 10.1007/978-3-319-02435-6. [26] Z.-Z. Sun, The stability and convergence of an explicit difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions,, J. Comput. Phys., 219 (2006), 879. doi: 10.1016/j.jcp.2006.07.001. [27] B. R. Wong, Numerical solution of the time-dependent Schrödinger equation,, in Frontiers in Physics, (2009), 396. [28] S.-S. Xie, G.-X. Li and S. Yi, Compact finite difference schemes with high accuracy for one-dimensional Schrödinger equation,, Comput. Meth. Appl. Mech. Engrg., 198 (2009), 1052. doi: 10.1016/j.cma.2008.11.011. [29] A. A. Zlotnik, Convergence rate estimates of finite-element methods for second-order hyperbolic equations,, in Numerical Methods and Applications (ed. G.I. Marchuk), (1994), 155. [30] A. A. Zlotnik, Error estimates of the Crank-Nicolson-polylinear FEM with the discrete TBC for the generalized Schrödinger equation in an unbounded parallelepiped,, in Finite Difference Methods, (2014), 18. [31] A. A. Zlotnik and A. V. Lapukhina, Stability of a Numerov type finite-difference scheme with approximate transparent boundary conditions for the nonstationary Schrödinger equation on the half-axis,, J. Math. Sci., 169 (2010), 84. doi: 10.1007/s10958-010-0040-9. [32] 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., 93 (2015), 279. doi: 10.1016/j.apnum.2014.05.003. [33] A. Zlotnik and I. Zlotnik, Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation,, Kinetic Relat. Models, 5 (2012), 639. doi: 10.3934/krm.2012.5.639. [34] A. Zlotnik and I. Zlotnik, The high order method with discrete TBCs for solving the Cauchy problem for the 1D Schrödinger equation,, Comput. Meth. Appl. Math., 15 (2015), 233. doi: 10.1515/cmam-2015-0007. [35] A. Zlotnik and I. Zlotnik, Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis,, Russ. J. Numer. Anal. Math. Model., 31 (2016).
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