Numerische Mathematik

, Volume 141, Issue 3, pp 681–691 | Cite as

Non-satisfiability of a positivity condition for commutator-free exponential integrators of order higher than four

  • Harald HofstätterEmail author
  • Othmar Koch


We consider commutator-free exponential integrators as put forward in Alverman and Fehske (J Comput Phys 230:5930–5956, 2011). For parabolic problems, it is important for the well-definedness that such an integrator satisfies a positivity condition such that essentially it only proceeds forward in time. We prove that this requirement implies maximal convergence order of four for real coefficients, which has been conjectured earlier by other authors.

Mathematics Subject Classification



Supplementary material


  1. 1.
    Alverman, A., Fehske, H.: High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys. 230, 5930–5956 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alverman, A., Fehske, H., Littlewood, P.: Numerical time propagation of quantum systems in radiation fields. New J. Phys. 14, 105008 (2012)CrossRefGoogle Scholar
  3. 3.
    Bader, P., Iserles, A., Kropielnicka, K., Singh, P.: Effective approximation for the linear time-dependent Schrödinger equation. Found. Comput. Math. 14, 689–720 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bader, P., Iserles, A., Kropielnicka, K., Singh, P.: Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential. Proc. R. Soc. A 472, 20150733 (2016)CrossRefzbMATHGoogle Scholar
  5. 5.
    Blanes, S., Casas, F., Chartier, P., Murua, A.: Optimized high-order splitting methods for some classes of parabolic equations. Math. Comp. 82, 1559–1576 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanes, S., Casas, F., Oteo, J., Ros, J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blanes, S., Casas, F., Thalhammer, M.: High-order commutator-free quasi-Magnus integrators for non-autonomous linear evolution equations. Comput. Phys. Commun. 220, 243–262 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Blanes, S., Casas, F., Thalhammer, M.: Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type. IMA J. Numer. Anal. 38, 743–778 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Blanes, S., Moan, P.: Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems. Appl. Numer. Math. 56, 1519–1537 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Castella, F., Chartier, P., Descombes, S., Vilmart, G.: Splitting methods with complex times for parabolic equations. BIT Numer. Math. 49, 487–508 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Celledoni, E.: Eulerian and semi-Lagrangian schemes based on commutator-free exponential integrators. Group Theory Numer. Anal. CRM Proc. Lecture Notes 39, 77–90 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Celledoni, E., Marthinsen, A., Owren, B.: Commutator-free Lie-group methods. Future Gen. Comput. Syst. 19(3), 341–352 (2003)CrossRefGoogle Scholar
  13. 13.
    Goldman, D., Kaper, T.: \(n\)th-order operator splitting schemes and nonreversible systems. SIAM J. Numer. Anal. 33(1), 349–367 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hansen, E., Ostermann, A.: High order splitting methods for analytic semigroups exist. BIT Numer. Math. 49, 527–542 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  16. 16.
    Iserles, A., Nørsett, S.: On the solution of linear differential equations on Lie groups. Phil. Trans. R. Soc. Lond. A 357, 983–1019 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Magnus, W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–000 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Owren, B.: Order conditions for commutator-free Lie group methods. J. Phys. A: Math. Gen. 39, 5585–5599 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Park, T., Light, J.: Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85, 5870–5876 (1986)CrossRefGoogle Scholar
  21. 21.
    Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209–228 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sheng, Q.: Solving linear partial differential equations by exponential splittings. IMA J. Numer. Anal. 9(2), 199–212 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WienWienAustria

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