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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:[email protected] (K. Kormann)
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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 

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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