Yun-an Yan. Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics[J]. Chinese Journal of Chemical Physics , 2017, 30(3): 277-286. doi: 10.1063/1674-0068/30/cjcp1703025
Citation: Yun-an Yan. Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics[J]. Chinese Journal of Chemical Physics , 2017, 30(3): 277-286. doi: 10.1063/1674-0068/30/cjcp1703025

Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics

doi: 10.1063/1674-0068/30/cjcp1703025
  • Received Date: 2017-03-03
  • Rev Recd Date: 2017-03-17
  • A wide range of quantum systems are time-invariant and the corresponding dynamics is dictated by linear differential equations with constant coefficients.Although simple in mathematical concept,the integration of these equations is usually complicated in practice for complex systems,where both the computational time and the memory storage become limiting factors.For this reason,low-storage Runge-Kutta methods become increasingly popular for the time integration.This work suggests a series of s-stage sth-order explicit RungeKutta methods specific for autonomous linear equations,which only requires two times of the memory storage for the state vector.We also introduce a 13-stage eighth-order scheme for autonomous linear equations,which has optimized stability region and is reduced to a fifth-order method for general equations.These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes.As an example,we apply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.


  • loading
  • [1]
    E. Schrödinger, Ann. Phys. 384, 361(1926).
    M. Born, W. Heisenberg, and P. Jordan, Zeit. Phys. 35, 557(1925).
    F. Bloch, Phys. Rev. 70, 460(1946).
    A. G. Redfield, IBM J. Res. Dev. 1, 19(1957).
    G. Lindblad, Commun. Math. Phys. 48, 119(1976).
    V. Shapiro and V. Loginov, Physica A 91, 563(1978).
    Y. Tanimura and R. Kubo, J. Phys. Soc. Jpn. 58, 101(1989).
    Y. Tanimura, Phys. Rev. A 41, 6676(1990).
    Y. A. Yan, F. Yang, Y. Liu, and J. Shao, Chem. Phys. Lett. 395, 216(2004).
    L. Adrianova, Introduction to Linear Systems of Diffierential Equations, Translations of Mathematical Monographs, Vol.146, Providence, Rhode Island:AMS, (1995).
    A. Greenbaum, Iterative Methods for Solving Linear Systems, Frontiers in Applied Mathematics, Vol.17, Philadelphia:SIAM, (1997).
    E. Coddington and R. Carlson, Linear Ordinary Diffierential Equations. Philadelphia:SIAM, (1997).
    A. Forsyth, Part 3. Ordinary Linear Equations, Theory of Diffierential Equations Vol.4, Cambridge:Cambridge University Press (1902).
    Y. Saad, Iterative Methods for Sparse Linear Systems 2nd Edn., Philadelphia:SIAM, (2003).
    C. Lanczos, J. Res. Nat. Bur. Standards 45, 255(1950).
    R. Haydock, Comput. Phys. Commun. 20, 11(1980).
    V. Druskin and L. Knizhnerman, J. Comput. Appl. Math. 50, 255(1994).
    H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81, 3967(1984).
    C. Leforestier, R. Bisseling, C. Cerjan, M. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. Meyer, N. Lipkin, O. Roncero, and R. Kosloff, J. Comput. Phys. 94, 59(1991).
    R. C. Ward, SIAM J. Numer. Anal. 14, 600(1977).
    M. Hochbruck, C. Lubich, and H. Selhofer, SIAM J. Sci. Comput. 19, 1552(1998).
    R. B. Sidje, ACM Trans. Math. Softw. 24, 130(1998).
    J. C. Butcher, Numerical Methods for Ordinary Diffierential Equations, 2nd Edn. Chichester:Wiley, (2008).
    R. E. Bellman, Dynamic Programming, Princeton:Princeton University Press, (1957).
    M. Feit, J. Fleck, and A. Steiger, J. Comput. Phys. 47, 412(1982).
    P. Houwen, Numer. Math. 20, 149(1972).
    P. J. Van Der Houwen, Construction of Integration Formulas for Initial Value Problems, Vol.19, Amsterdam:North-Holland, (1977).
    A. Wray, NASA Ames Research Center, California:Moffett Field, 202(1990).
    J. Williamson, J. Comput. Phys. 35, 48(1980).
    S. Gill, Math. Proc. Cambridge Phil. Soc. 47, 96(1951).
    E. K. Blum, Math. Comput. 16, 176(1962).
    D. J. Fyfe, Math. Comput. 20, 392(1966).
    D. I. Ketcheson, SIAM J. Sci. Comput. 30, 2113(2008).
    M. H. Carpenter and C. A. Kennedy, Fourth-order 2Nstorage Runge-Kutta schemes, Tech. Rep. NASATM-109112, VA:National Aeronautics and Space Administration, NASA Langley Research Center, (1994).
    C. A. Kennedy, M. H. Carpenter, and R. Lewis, Appl. Numer. Math. 35, 177(2000).
    J. Butcher, BIT Numer. Math. 25, 521(1985).
    J. Niegemann, R. Diehl, and K. Busch, J. Comput. Phys. 231, 364(2012).
    T. Becker, H. Kredel, and V. Weispfenning, Gröbner Bases:a Computational Approach to Commutative Algebra, Corrected Edn., Graduate Texts in Mathematics, Vol.141, London, UK:Springer-Verlag, (1993).
    D. Zingg and T. Chisholm, Appl. Numer. Math. 31, 227(1999).
    J. A. Nelder and R. Mead, Comput. J. 7, 308(1965).
    S. Johnson, The Nlopt Nonlinear-Optimization Package.
    K. Levenberg, Quart. J. Appl. Math. 2, 164(1944).
    D. W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431(1963).
    M. Lourakis, Levmar:Levenberg-Marquardt Nonlinear Least Squares Algorithms in C/C++, (2004).
    E. Fehlberg, Computing 6, 61(1970).
    M. Calvo, J. Montijano, and L. Randez, Comput. Math. Appl. 20, 15(1990).
    P. Prince and J. Dormand, J. Comput. Appl. Math. 7, 67(1981).
    C. W. Tang, Appl. Phys. Lett. 48, 183(1986).
    H. Lee, Y. C. Cheng, and G. R. Fleming, Science 316, 1462(2007).
    J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, Nature 347, 539(1990).
    G. Trinkunas, J. L. Herek, T. Polıvka, V. Sundström, and T. Pullerits, Phys. Rev. Lett. 86, 4167(2001).
    V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, 3rd Edn. Weinheim:WILEY-VCH, (2010).
    M. del Rey, A. W. Chin, S. F. Huelga, and M. B. Plenio, J. Phys. Chem. Lett. 4, 903(2013).
    Z. Chen, V. Stepanenko, V. Dehm, P. Prins, L. Siebbeles, J. Seibt, P. Marquetand, V. Engel, and F. Wrthner, Chem. Eur. J. 13, 436(2007).
    X. Q. Li, X. Zhang, S. Ghosh, and F. Würthner, Chem. Eur. J. 14, 8074(2008).
    M. R. Wasielewski, Acc. Chem. Res. 42, 1910(2009).
    S.Wolter, J. Aizezers, F. Fennel, M. Seidel, F.Wurthner, O. Kühn, and S. Lochbrunner, New J. Phys. 14, 105027(2012).
    H. Marciniak, X. Q. Li, F. Würthner, and S. Lochbrunner, J. Phys. Chem. A 115, 648(2011).
    D. Ambrosek, H. Marciniak, S. Lochbrunner, J. Tatchen, X. Li, F. Würthner, and O. Kühn, Phys. Chem. Chem. Phys. 13, 17649(2011).
    D. Ambrosek, A. Köhn, J. Schulze, and O. Kühn, J. Phys. Chem. A 116, 11451(2012).
    Q. Shi, L. Chen, G. Nan, R. X. Xu, and Y. Yan, J. Chem. Phys. 130, 084105(2009).
    Y. A. Yan and O. Kühn, New J. Phys. 14, 105004(2012).
    Y. A. Yan and S. Cai, J. Chem. Phys. 141, 054105(2014).
    Y. Yan, J. Chem. Phys. 144, 024305(2016).
    O. Kühn and V. Sundström, J. Chem. Phys. 107, 4154(1997).
    H. Grabert, P. Schramm, and G. L. Ingold, Phys. Rep. 168, 115(1988).
    J. Shao, J. Chem. Phys. 120, 5053(2004).
    M. Schröter, S. Ivanov, J. Schulze, S. Polyutov, Y. Yan, T. Pullerits, and O. Kühn, Phys. Rep. 567, 1(2015).
    Y. Yan and J. Shao, Front. Phys. 11, 110309(2016).
    Y. A. Yan and Y. Zhou, Hybrid Stochastic-Hierarchical Equations. (2012).
  • 加载中


    通讯作者: 陈斌,
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1271) PDF downloads(825) Cited by()
    Proportional views


    DownLoad:  Full-Size Img  PowerPoint