Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T21:19:48.894Z Has data issue: false hasContentIssue false
  • English
  • Français

A Time-Space Adaptive Method for the Schrödinger Equation

Part of: Quantum theory Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  22 June 2016

Katharina Kormann
Katharina Kormann*
Affiliation:
Technische Universität München, Zentrum Mathematik, Boltzmannstr. 3, 85747 Garching, Germany
*
*Corresponding author. Email address:katharina.kormann@tum.de (K. Kormann)
Get access

Abstract

In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.

Keywords

a posteriori error estimateadaptive mesh refinementfinite/spectral elementstime-dependent Schrödinger equation

MSC classification

Secondary: 65M50: Mesh generation and refinement 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65Z05: Applications to physics 81-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 60 - 85
Copyright
Copyright © Global-Science Press 2016 

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

[1] 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. Comput. Phys., 4:726–796, 2008.Google Scholar
[2] Babuška, I. and Miller, A. D.. The post-processing approach in the finite element method, III: A posteriori error estimation and adaptive mesh selection. Int. J. Numer. Meth. Eng., 20:2311–2324, 1984.CrossRefGoogle Scholar
[3] Babuška, I. and Rheinboldt, W. C.. A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng., 12:1597–1615, 1978.CrossRefGoogle Scholar
[4] Babuška, I. and Rheinboldt, W. C.. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15:736–754, 1978.CrossRefGoogle Scholar
[5] Bangerth, W., Hartmann, R., and Kanschat, G.. deal.II — a general-purpose object-oriented finite element library. ACM Trans. Math. Softw., 33(4), 2007.CrossRefGoogle Scholar
[6] Bangerth, W., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., and Young, T. D.. The deal.II library, version 8.1. arXiv preprint, 2013.Google Scholar
[7] Bangerth, W. and Kayser-Herold, O.. Data structures and requirements for hp finite element software. ACM Trans. Math. Softw., 36(1):4/1–4/31, 2009.CrossRefGoogle Scholar
[8] Becker, R. and Rannacher, R.. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math., 4:237–264, 1996.Google Scholar
[9] Becker, R. and Rannacher, R.. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10:1–102, 2001.CrossRefGoogle Scholar
[10] Ben-Nun, M., Quenneville, J., and Martínez, T. J.. Ab initio multiple spawning: Photochemistry from first principles quantum molecular dynamics. J. Phys. Chem. A, 104:5161–5175, 2000.CrossRefGoogle Scholar
[11] Bermejo, R. and Carpio, J.. A space-time adaptive finite element algorithm based on dual weighted residual methodology for parabolic equations. SIAM J. Sci. Comput., 31(5):3324–3355, 2009.CrossRefGoogle Scholar
[12] Billing, G. D.. Time-dependent quantum dynamics in a Gauss–Hermite basis. J. Chem. Phys., 110:5526–5537, 1998.CrossRefGoogle Scholar
[13] Blanes, S., Casas, F., Oteo, J., and Ros, J.. The Magnus expansion and some of its applications. Phys. Rep., 470:151–238, 2009.CrossRefGoogle Scholar
[14] Cao, Y. and Petzold, L.. A posteriori error estimate and global error control for ordinary differential equations by the adjoint method. SIAM J. Sci. Comput., 26(3):359–374, 2004.CrossRefGoogle Scholar
[15] Cohen, G. C.. Higher-Order Numerical Methods for Transient Wave Equations. Springer Verlag, Berlin, 2002.CrossRefGoogle Scholar
[16] Dörfler, W.. A time- and spaceadaptive algorithm for the linear time-dependent Schrödinger equation. Numer. Math., 73:419–448, 1996.Google Scholar
[17] Durufle, M., Grob, P., and Joly, P.. Influence of Gauss and Gauss–Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain. Numer. Meth. Part. D. E., 25:526–551, 2009.CrossRefGoogle Scholar
[18] Eriksson, K. and Johnson, C.. Adaptive finite element methods for parabolic problem. I: A linear model problem. SIAM J. Numer. Anal., 28:43–77, 1991.CrossRefGoogle Scholar
[19] Faou, E. and Gradinaru, V.. Gauss-Hermite wave packet dynamics: convergence of the spectral and pseudo-spectral approximation. IMA J. Numer. Anal., 29:1023–1045, 2009.CrossRefGoogle Scholar
[20] Faou, E., Gradinaru, V., and Lubich, C.. Computing semiclassical quantum dynamics with Hagedorn wavepackets. SIAM J. Sci. Comput., 31:3027–3041, 2009.CrossRefGoogle Scholar
[21] Fattal, E., Baer, R., and Kosloff, R.. Phase space approach for optimizing grid representations: The mapped Fourier method. Phys. Rev. E, 53:1217–1227, 1996.CrossRefGoogle ScholarPubMed
[22] Feit, M. D., Fleck, J., J. A., , and Steiger, A.. Solution of the Schrödinger equation by a spectral method. J. Comput. Phys., 47:412–433, 1982.CrossRefGoogle Scholar
[23] Gagelman, J. and Yserentant, H.. A spectral method for Schrödinger equations with smooth confinement potentials. Numer. Math., 122:383–398, 2012.CrossRefGoogle Scholar
[24] Gradinaru, V.. Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation. Computing, 80:1–22, 2007.CrossRefGoogle Scholar
[25] Griebel, M. and Hamaekers, J.. Sparse grids for the Schrödinger equation. ESAIM-Math. Model Num., 41:215–247, 2007.CrossRefGoogle Scholar
[26] Gustafsson, B.. High Order Difference Methods for Time Dependent PDE. Springer, Berlin, 2008.Google Scholar
[27] Gustafsson, M. and Holmgren, S.. An implementation framework for solving high-dimensional PDEs on massively parallel computers. In Kreiss, G., Lötstedt, P., Målqvist, A., and Neytcheva, M., editors, Numerical Mathematics and Advanced Applications 2009, pages 417–424. Springer, Berlin, 2010.CrossRefGoogle Scholar
[28] Hallatschek, K.. Fouriertransformation auf dünnen Gittern mit hierarchischen Basen. Numer. Math., 63:83–97, 1992.CrossRefGoogle Scholar
[29] Han, H., Yin, D., and Huang, Z.. Numerical solutions of Schrödinger equations in r 3 . Numer. Meth. Partial Diff. Eqs., 23:511–533, 2007.CrossRefGoogle Scholar
[30] Heller, E. J.. Frozen Gaussians: A very simple semiclassical approximation. J. Chem. Phys., 75:2923–2931, 1981.CrossRefGoogle Scholar
[31] Hochbruck, M. and Lubich, C.. On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal., 41:945–963, 2003.CrossRefGoogle Scholar
[32] Hochbruck, M., Lubich, C., and Selhofer, H.. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5), 1999.Google Scholar
[33] Karakashian, O. and Makridakis, C.. A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method. SIAM J. Numer. Anal., 36:1779–1807, 1999.CrossRefGoogle Scholar
[34] Karniadakis, G. E. and Sherwin, S. J.. Spectral/hp element methods for computational fluid dynamics. Oxford University Press, 2nd edition, 2005.CrossRefGoogle Scholar
[35] Kleinekathöfer, U. and Tannor, D. J.. Extension of the mapped Fourier method to time-dependent problems. Phys. Rev. E, 60:4926, 1999.CrossRefGoogle ScholarPubMed
[36] Kormann, K., Holmgren, S., and Karlsson, H. O.. Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. J. Chem. Phys., 128:184101, 2008.CrossRefGoogle ScholarPubMed
[37] Kormann, K., Holmgren, S., and Karlsson, H. O.. Global error control of the progagation for the Schrödinger equation with a time-dependent Hamiltonian. J. Comput. Sci., 2:178–187, 2011.CrossRefGoogle Scholar
[38] Kormann, K. and Kronbichler, M.. Parallel finite element operator application: Graph partitioning and coloring. In 2011 Seventh IEEE International Conference on eScience, pages 332–339, 2011.CrossRefGoogle Scholar
[39] Kormann, K. and Nissen, A.. Error control for simulations of a dissociative quantum system. In Kreiss, G., Lötstedt, P., Målqvist, A., and Neytcheva, M., editors, Numerical Mathematics and Advanced Applications 2009, pages 523–531. Springer, Berlin, 2010.CrossRefGoogle Scholar
[40] Kosloff, D. and Kosloff, R.. A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics. J. Comput. Phys., 52:35–53, 1983.CrossRefGoogle Scholar
[41] Kronbichler, M. and Kormann, K.. A generic interface for parallel cell-based finite element operator application. Comp. Fluids, 63:135–147, 2012.CrossRefGoogle Scholar
[42] Kyza, I.. A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations. ESAIM-Math. Model. Num., 45:761–778, 2011.CrossRefGoogle Scholar
[43] Lubich, C.. Integrators for Quantum Dynamics: A Numerical Analyst's Brief Review. In Grotendorst, J., Marx, D., and Muramatsu, A., editors, Quantum Simulation of Complex Many-Body Systems: From Theory to Algorithms, pages 459–466. John von Neumann Institute for Computing, 2002.Google Scholar
[44] Manolopoulos, D. E. and Wyatt, R. E.. Quantum scattering via the log derivative version of the Kohn variational principle. Chem. Phys. Lett., 152:23–32, 1988.CrossRefGoogle Scholar
[45] H.-D.|Meyer, Gatti, F., and G. W., (Eds.). Multidimensional Quantum Dynamics: MCTDH Theory and Applications. Wiley-VCH, Weinheim, 2009.Google Scholar
[46] Miller, W. H.. Semiclassical methods in chemical physics. Science, 233:171–177, 1986.CrossRefGoogle ScholarPubMed
[47] Park, T. J. and Light, J. C.. Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys., 85(10), 1986.CrossRefGoogle Scholar
[48] Picasso, M.. Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Eng., 167:223–237, 1998.CrossRefGoogle Scholar
[49] Rescigno, T. N. and McCurdy, C. W.. Numerical grid methods for quantum-mechanical scattering problems. Phys. Rev. A, 62:032706, 2000.CrossRefGoogle Scholar
[50] Rheinboldt, W. C. and Mesztenyi, C. K.. On a Data Structure for Adaptive Finite Element Mesh Refinements. ACM Trans. Math. Softw., 6:166–187, 1980.CrossRefGoogle Scholar
[51] Saad, Y.. Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal., 29:209–228, 1992.CrossRefGoogle Scholar
[52] Schmich, M. and Vexler, B.. Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput., 30(1):369–393, 2008.CrossRefGoogle Scholar
[53] Sun, Z., Lou, N., and Nyman, G.. Time-Dependent Wave Packet Split Operator Calculations on a Three-Dimensional Fourier Grid in Radau Coordinates Applied to the OClO Photoelectron Spectrum. J. Phys. Chem. A, 108:9226–9232, 2004.CrossRefGoogle Scholar
[54] Thomée, V.. Galerkin Finite Element Methods for Parabolic Problems (2nd edn.), volume 25 of Springer Series in Computational Mathematics. Springer, Berlin, 2006.Google Scholar
[55] A.|Walther. Program Reversal Schedules for Single and Multi-processor Machines. PhD thesis, Institute of Scientific Computing, TU Dresden, 1999.Google Scholar
[56] Weideman, J. A. C.. Spectral differentiation matrices for the numerical solution of Schrödinger's equation. J. Phys. A: Math. Gen., 39:10229–10237, 2006.CrossRefGoogle Scholar
[57] Wu, Y. and Batista, V. S.. Matching-pursuit for simulations of quantum processes. J. Chem. Phys., 118:6720–6724, 2003.CrossRefGoogle Scholar
[58] Zewail, A. H.. Laser femtochemistry. Science, 242:1645–1653, 1988.CrossRefGoogle ScholarPubMed