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Error analysis of high-order splitting methodsfor nonlinear evolutionary Schrödinger equationsand application to the MCTDHF equationsin electron dynamics

Published online by Cambridge University Press:  09 July 2013

Othmar Koch ,
Christof Neuhauser  and
Mechthild Thalhammer
Othmar Koch
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria.. othmar@othmar-koch.org
Christof Neuhauser
Affiliation:
Institut für Mathematik, Leopold–Franzens Universität Innsbruck, Technikerstrasse 13/VII, 6020 Innsbruck, Austria.; christof.neuhauser@uibk.ac.at
Mechthild Thalhammer
Affiliation:
Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, Werner–Heisenberg–Weg 39, 85577 Neubiberg, Germany.; mechthild.thalhammer@uibk.ac.at
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Abstract

In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (nonstiff) order p applied to the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations, which are associated with a model reduction for high-dimensional linear Schrödinger equations describing free electrons that interact by Coulomb force. Provided that the analytical solution of the MCTDHF equations constituting a system of coupled linear ordinary differential equations and low-dimensional nonlinear partial differential equations satisfies suitable regularity requirements, convergence of order p − 1 in the H1 Sobolev norm and convergence of order p in the L2 norm is proven. An analogous result follows for the cubic nonlinear Schrödinger equation, which is also illustrated by a numerical experiment.

Keywords

Nonlinear evolution equationstime-dependent nonlinear Schrödinger equationsmulti-configuration time-dependent Hartree–Fock (MCTDHF) equationshigh-order exponential operator splitting methodslocal error expansionconvergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1265 - 1286
Copyright
© EDP Sciences, SMAI, 2013

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References

Abhau, J. and Thalhammer, M., A numerical study of adaptive space and time discretisations for Gross–Pitaevskii equations. J. Comput. Phys. 231 (2012) 6665–6681. Google Scholar
R.A. Adams, Sobolev Spaces. Academic Press, Orlando, Fla. (1975).
Bao, W., Jaksch, D. and Markowich, P., Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comput. Phys. 187 (2003) 318–342. Google Scholar
Bao, W. and Shen, J., A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates. SIAM J. Sci. Comput. 26 (2005) 2010–2028. Google Scholar
Bardos, C., Catto, I., Mauser, N. and Trabelsi, S., Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree-Fock equations: A sufficient condition. Appl. Math. Lett. 22 (2009) 147–152. Google Scholar
Bardos, C., Catto, I., Mauser, N. and Trabelsi, S., Setting and analysis of the multi-configuration time-dependent Hartree–Fock equations. Arch. Ration. Mech. Anal. 198 (2010) 273–330. Google Scholar
Beck, M.H., Jäckle, A., Worth, G.A., and Meyer, H.-D., The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets. Phys. Rep. 324 (2000) 1–105. Google Scholar
Beck, M.H. and Meyer, H.-D., An efficient and robust integration scheme for the equations of the multiconfiguration time-dependent Hartree (MCTDH) method. Z. Phys. D 42 (1997) 113–129. Google Scholar
Blanes, S. and Moan, P.C., Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142 (2002) 313–330. Google Scholar
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer Verlag, New York, 2nd edition (2002).
Burghardt, I., Meyer, H.-D. and Cederbaum, L.S., Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time-dependent Hartree method. J. Chem. Phys. 111 (1999) 2927–2939. Google Scholar
Caillat, J., Zanghellini, J., Kitzler, M., Kreuzer, W., Koch, O. and Scrinzi, A., Correlated multielectron systems in strong laser pulses – an MCTDHF approach. Phys. Rev. A 71 (2005) 012712. Google Scholar
Caliari, M., Neuhauser, Ch. and Thalhammer, M., High-order time-splitting Hermite and Fourier spectral methods for the Gross–Pitaevskii equation. J. Comput. Phys. 228 (2009) 822–832. Google Scholar
Descombes, S. and Thalhammer, M., An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT Numer. Math. 50 (2010) 729–749. Google Scholar
Dirac, P.A.M., Note on exchange phenomena in the Thomas atom. Proc. Cambridge Philos. Soc. 26 (1930) 376–385. Google Scholar
J. Frenkel, Wave Mechanics, Advanced General Theory. Clarendon Press, Oxford (1934).
Gauckler, L., Convergence of a split-step Hermite method for the Gross–Pitaevskii equation. IMA J. Numer. Anal. 49 (2011) 1194–1209. Google Scholar
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Springer Verlag, Berlin–Heidelberg–New York (2002).
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Springer Verlag, Berlin–Heidelberg–New York (1987).
G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities. Cambridge Univ. Press, Cambridge (1934).
T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Berlin–Heidelberg–New York (1966).
Kato, T. and Kono, H., time-dependent multiconfiguration theory for electronic dynamics of molecules in an intense laser field. Chem. Phys. Lett. 392 (2004) 533–540. Google Scholar
Kitzler, M., Zanghellini, J., Jungreuthmayer, Ch., Smits, M., Scrinzi, A. and Brabec, T., Ionization dynamics of extended multielectron systems. Phys. Rev. A 70 (2004) 041401(R). Google Scholar
Koch, O., The variational splitting method for the multi-configuration time-dependent Hartree–Fock equations for atoms. To appear in J. Numer. Anal. Indust. Appl. Math. 7 (2012) 1–13. Google Scholar
O. Koch, W. Kreuzer and A. Scrinzi, MCTDHF in ultrafast laser dynamics. AURORA TR-2003-29, Inst. Appl. Math. Numer. Anal., Vienna Univ. of Technology, Austria (2003). Available at http://www.othmar-koch.org/research.html.
Koch, O., Kreuzer, W. and Scrinzi, A., Approximation of the time-dependent electronic Schrödinger equation by MCTDHF. Appl. Math. Comput. 173 (2006) 960–976. Google Scholar
Koch, O. and Lubich, C., Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics. M2AN Math. Model. Numer. Anal. 41 (2007) 315–331. Google Scholar
O. Koch and C. Lubich, Analysis and time integration of the multi-configuration time-dependent Hartree–Fock equations in electron dynamics. ASC Report 4/2008, Inst. Anal. Sci. Comput. Vienna Univ. of Technology (2008).
Koch, O. and Lubich, C., Variational splitting time integration of the MCTDHF equations in electron dynamics. IMA J. Numer. Anal. 31 (2011) 379–395. Google Scholar
Kwon, Y., Ceperley, D.M. and Martin, R.M., Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study. Phys. Rev. B 58 (1998) 6800–6806. Google Scholar
L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, Oxford–New York, 3rd edition (1977).
Lubich, C., A variational splitting integrator for quantum molecular dynamics. Appl. Numer. Math. 48 (2004) 355–368. Google Scholar
C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. Zurich Lect. Adv. Math. Europ. Math. Soc., Zurich (2008).
Lubich, C., On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77 (2008) 2141–2153. Google Scholar
McLachlan, R. and Quispel, R., Splitting methods. Acta Numer. 11 (2002) 341–434. Google Scholar
H.-D. Meyer, F. Gatti and G.A. Worth, editors. Multidimensional Quantum Dynamics: MCTDH Theory and Applications. Wiley-VCH, Weinheim, Berlin (2009).
Meyer, H.-D., Manthe, U. and Cederbaum, L.S., The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165 (1990) 73–78. Google Scholar
Meyer, H.-D. and Worth, G.A., Quantum molecular dynamics: Propagating wavepackets and density operators using the multi-configuration time-dependent Hartree (MCTDH) method. Theo. Chem. Acc. 109 (2003) 251–267. Google Scholar
M. Miklavčič, Applied Functional Analysis and Partial Differential Equations. World Scientific, Singapore (1998).
Nagy, I., Diez Muiño, R., Juaristi, J.I. and Echenique, P.M., Spin-resolved pair-distribution functions in an electron gas: A scattering approach based on consistent potentials. Phys. Rev. B 69 (2004) 233105. Google Scholar
Nest, M. and Klamroth, T., Correlated many-electron dynamics: Application to inelastic electron scattering at a metal film. Phys. Rev. A 72 (2005) 012710. Google Scholar
Nest, M., Klamroth, T. and Saalfrank, P., The multiconfiguration time-dependent Hartree–Fock method for quantum chemical calculations. J. Chem. Phys. 122 (2005) 124102. Google ScholarPubMed
Neuhauser, C. and Thalhammer, M., On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numer. Math. 49 (2009) 199–215. Google Scholar
Perez-Garcia, V.M. and Liu, X., Numerical methods for the simulation of trapped nonlinear Schrödinger systems. Appl. Math. Comput. 144 (2003) 215–235. Google Scholar
J.C. Slater, Quantum Theory of Molecules and Solids. McGraw–Hill, New York, Toronto, London 1 (1960).
Strang, G., On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506–517. Google Scholar
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Appl. Math. Sci. Springer Verlag, New York (1999).
Thalhammer, M., High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46 (2008) 2022–2038. Google Scholar
Thalhammer, M., Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2012) 3231–3258. Google Scholar
Trabelsi, S., Solutions of the multiconfiguration time-dependent Hartree–Fock equations with Coulomb interactions. C. R. Acad. Sci. Paris, Ser. I 345 (2007) 145–150. Google Scholar
Trotter, H.F., On the product of semi-groups of operators. Proc. Amer. Math. Soc. 10 (1959) 545–551. Google Scholar
Zanghellini, J., Kitzler, M., Brabec, T. and Scrinzi, A., Testing the multi-configuration time-dependent Hartree–Fock method. J. Phys. B: At. Mol. Phys. 37 (2004) 763–773. Google Scholar
Zanghellini, J., Kitzler, M., Fabian, C., Brabec, T. and Scrinzi, A., An MCTDHF approach to multi-electron dynamics in laser fields. Laser Phy. 13 (2003) 1064–1068. Google Scholar