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Multi-Product Expansion with Suzuki’s Method: Generalization

Published online by Cambridge University Press:  28 May 2015

Jürgen Geiser
Jürgen Geiser*
Affiliation:
Department of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin, Germany
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Abstract

In this paper we discuss the extension to exponential splitting methods with respect to time-dependent operators. We concentrate on the Suzuki’s method, which incorporates ideas to the time-ordered exponential of [3,11,12,34]. We formulate the methods with respect to higher order by using kernels for an extrapolation scheme. The advantages include more accurate and less computational intensive schemes to special time-dependent harmonic oscillator problems. The benefits of the higher order kernels are given on different numerical examples.

Keywords

65M1565L0565M71Suzuki’s methodmultiproduct expansionmultiplicate and additive splitting schemesexponential splitting
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 1 , February 2011 , pp. 68 - 91
Copyright
Copyright © Global Science Press Limited 2011

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References

[1] Aguilera-Navarro, VC., Estévez, G.A. and Guardiola, R., Variational andperturbative schemes for a spiked harmonic oscillator, J. Math. Phys. 31, 99 (1990).Google Scholar
[2] Baye, D., Goldstein, G. and Capel, P., Fourth-order factorization of the evolution operator for time-dependent potentials, Phys. Letts, A 317, 337, 2003.Google Scholar
[3] Goldstein, G. and Baye, D., Sixth-order factorization of the evolution operator for time-dependent potentials, Phys. Rev, E 70, 056703, 2004.Google Scholar
[4] Buendía, E., Gálvez, F.J., and Puertas, A., Study of the singular anharmonic potentials by means of the analytic continuation method, J. Phys. A 28, 6731 (1995).Google Scholar
[5] Blanes, S. and Moan, P C., Practical symplectic partition Runge-Kutta methods and Runge-Kutta Nyström methods, J. Comput. Appl. Math. 142, 3131 (2002).Google Scholar
[6] Blanes, S., Casas, F. and Ros, J., Extrapolation of Symplectic Integrators, Applied Numerical Mathematics, 56, 1519–1537, 2006.Google Scholar
[7] Blanes, S. and Moan, P.C., Fourth- and sixt-order commutator free Magnus integrators for, linear and nonlinear dynamical systems, Applied Numerical Mathematics, 56, (2006), pp. 1519–1537.Google Scholar
[8] Blanes, S. and Moan, P.C., Splitting Methods for Non-autonomous Hamiltonian Equations, Journal of Computational Physics, 170, (2001), pp. 205–230.Google Scholar
[9] Blanes, S. and Casas, F., Splitting Methods for Non-autonomous separabel dynamical systems, Journal of Physics A: Math. Gen., 39, (2006), pp. 5405–5423.CrossRefGoogle Scholar
[10] Blanes, S., Casas, F., Oteo, J.A. and Ros, J., The Magnus expansion and some of its applications, http://www.citebase.org/abstract?id=oai:arXiv.org:0810.5488, 2008.Google Scholar
[11] Chin, S.A. and Chen, C.R., Gradient symplectic algorithms for solving the SchrÃűdinger equation with time-dependent potentials, Journal of Chemical Physics, Vol. 117, no. 4, (2002), 1409–1415.Google Scholar
[12] Chin, S.A. and Anisimov, P., Gradient Symplectic Algorithms f or Solving the Radial Schrödinger Equation, J. Chem. Phys. 124, 054106, 2006.CrossRefGoogle ScholarPubMed
[13] Chin, S.A., Multi-product splitting and Runge-Kutta-Nyström integrators, Applied Numerical Mathematics, under review, 2008.Google Scholar
[14] Chin, S., Janecek, S., and Krotscheck, E., Any order imaginary time propagation method for solving the Schrödinger equation, Chemical Physics Letters, v. 470, iss. 4-6, (2009), pp. 342–346.Google Scholar
[15] Dyson, F.J., The radiation theorem ofTomonaga, Swinger and Feynman, Phys. Rev., 75, (1976), pp. 486–502.Google Scholar
[16] Farago, I., Geiser, J., Iterative Operator-Splitting Methods for Linear Problems, Preprint No. 1043 of the Weierstass Institute for Applied Analysis and Stochastics, (2005) 1–18. International Journal of Computational Science and Engineering, accepted September 2007.Google Scholar
[17] Forest, E. and Ruth, R. D., 4th-order symplectic integration, Physica D 43, 105 (1990).Google Scholar
[18] Geiser, J., Fourth-order splitting methods for time-dependant differential equations, Numer. Math. Theor. Meth. Appl., 1, (2008), pp. 321–339.Google Scholar
[19] Geiser, J. and Chin, S., Multi-product expansion, Suzuki’s method and the Magnus integrator for solving time-dependent problems, Preprint 2009-4, Humboldt University of Berlin, Department of Mathematics, Germany, 2009.Google Scholar
[20] Hairer, E., Norsett, S.P. and Wanner, G., Solving Ordinary Differential Equations I - Nonstiff Problems, Second Edition, Springer Verlag, Berlin, 1993.Google Scholar
[21] Hochbruck, M. and Ostermann, A., Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems, SIAM Journal on Numerical Analyis, Vol. 43, Iss. 3, (2005), pp. 1069–1090.Google Scholar
[22] Hansen, E. and Ostermann, A., Exponential splitting for unbounded operators, Math. Comp., 78, (2009), pp. 1485–1496.Google Scholar
[23] Jahnke, T. and Lubich, C. Error bounds for exponential operator splittings, BIT Numerical Mathematics, vol.40, no.4, (2000), pp. 735–745.CrossRefGoogle Scholar
[24] Killingbeck, J., Direct expectation value calculations, J. Phys. A 18, 245 (1985).Google Scholar
[25] Killingbeck, J.P., Jolicard, G., and Grosjean, A., A simple numerical method for singular potentials, J. Phys. A 34, L367 (2001).Google Scholar
[26] Mclachlan, R.I., On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comput. 16, 151 (1995).CrossRefGoogle Scholar
[27] Moan, P.C. and Niesen, J., Convergence of the exponential Lie series, Technical report, La Trobe University, September 2006.Google Scholar
[28] Moan, P.C. and Niesen, J., Convergence of the Magnus series, J. Found. of Comp. Math., vol.8, no. 3, (2008), pp. 291–301.Google Scholar
[29] Paul Raj, S., Rajasekar, S. and Murali, K.. Coexisting chaotic attractors, their basin of attractions and synchronization of chaos in two coupled Duffing oscillators, Physics Letters A, 264(4): 283–288, 1999.Google Scholar
[30] Roy, A.K., Calculation of the spiked harmonic oscillators through a generalized pseudospectral method, Phys. Lett. A 321 , 231 (2004).Google Scholar
[31] Sidi, A., Practical Extrapolation Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2003.Google Scholar
[32] Strang, G., On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5, (1968), 506–517.CrossRefGoogle Scholar
[33] Yoshida, K., Functional Analysis, Classics in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 1980.Google Scholar
[34] Suzuki, M., General decomposition theory of ordered exponentials, Proc. Japan Acad., Vol. 69, Ser. B, 161, (1993).Google Scholar
[35] Thalhammer, M., Higher-order splitting for Schrödinger equations, SIAM J. Numer. Anal., Vol. 46, No. 4, (2008), pp. 2022–2038.Google Scholar
[36] Wiebe, N., Berry, D.W., Hoyer, P. and Sanders, B.C.. Higher order decompositions of ordered operator exponentials, arXiv.org:0812.0562 (2008).Google Scholar