Skip to main content
SpringerLink
Log in

Splitting methods in the numerical integration of non-autonomous dynamical systems

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

We present a procedure leading to efficient splitting schemes for the time integration of explicitly time dependent partitioned linear differential equations arising when certain partial differential equations are previously discretized in space. In the first stage we analyze the order conditions of the corresponding autonomous problem and construct new 6th-order methods. In the second stage, by following a procedure previously designed by the authors, we generalize the methods to the time dependent case in such a way that no order reduction is present. The resulting schemes compare favorably with other integrators previously available.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blanes S., Casas F.: Splitting methods for non-autonomous separable dynamical systems. J. Phys. A. Math. Gen. 39, 5405–5423 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blanes S., Casas F., Murua A.: Symplectic splitting operator methods tailored for the time-dependent Schrödinger equation. J. Chem. Phys. 124, 234105 (2006)

    Article  Google Scholar 

  3. Blanes S., Casas F., Murua A.: Splitting methods for non-autonomous linear systems. Int. J. Comput. Math. 84, 713–727 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blanes S., Casas F., Murua A.: On the linear stability of splitting methods. Found. Comp. Math. 8, 357–393 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blanes S., Casas F., Murua A.: Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Math. Apl. 45, 87–143 (2008)

    Google Scholar 

  6. Blanes, S., Casas, F., Murua, A.: Error analysis of splitting methods for the time dependent Schrödinger equation. arXiv:1001.1549 (2011)

  7. Blanes S., Casas F., Oteo J.A., Ros J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)

    Article  MathSciNet  Google Scholar 

  8. Blanes S., Casas F., Ros J.: Improved high order integrators based on Magnus expansion. BIT 40, 434–450 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Blanes S., Diele F., Marangi C., Ragni S.: Splitting and composition methods for explicit time dependence in separable dynamical systems. J. Comput. Appl. Math. 235, 646–659 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Blanes S., Moan P.C.: Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142, 313–330 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gray S., Manolopoulos D.E.: Symplectic integrators tailored to the time-dependent Schrödinger equation. J. Chem. Phys. 104, 7099–7112 (1996)

    Article  Google Scholar 

  12. Gray S., Verosky J.M.: Classical Hamiltonian structures in wave packet dynamics. J. Chem. Phys. 100, 5011–5022 (1994)

    Article  Google Scholar 

  13. Hairer E., Lubich C., Wanner G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd ed. Springer, Berlin (2006)

    MATH  Google Scholar 

  14. Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A.: Lie group methods. Acta Numer. 9, 215–365 (2000)

    Article  Google Scholar 

  15. Leimkuhler B., Reich S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  16. Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  17. McLachlan R.I, Quispel R.: Splitting methods. Acta Numer. 11, 341–434 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. McLachlan R.I, Quispel R.G.W.: Geometric integrators for ODEs. J. Phys. A. Math. Gen. 39, 5251–5285 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rieben R., White D., Rodrigue G.: High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations. IEEE Trans. Antennas Propag. 52, 2190–2195 (2004)

    Article  MathSciNet  Google Scholar 

  20. Sanz-Serna J.M., Calvo M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)

    MATH  Google Scholar 

  21. Sanz-Serna J.M., Portillo A.: Classical numerical integrators for wave-packet dynamics. J. Chem. Phys. 104, 2349–2355 (1996)

    Article  Google Scholar 

  22. Sofroniou M., Spaletta G.: Derivation of symmetric composition constants for symmetric integrators. Optim. Methods Softw. 20, 597–613 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Walker R.B., Preston K.: Quantum versus classical dynamics in treatment of multiple photon excitation of anharmonic-oscillator. J. Chem. Phys. 67, 2017–2028 (1977)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022, Valencia, Spain

    Sergio Blanes

  2. Departament de Matemàtiques, Institut de Matemàtiques i Aplicacions de Castelló, Universitat Jaume I, 12071, Castellón, Spain

    Fernando Casas

  3. Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, EHU/UPV, Donostia/San Sebastián, Spain

    Ander Murua

Authors
  1. Sergio Blanes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ander Murua
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fernando Casas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blanes, S., Casas, F. & Murua, A. Splitting methods in the numerical integration of non-autonomous dynamical systems. RACSAM 106, 49–66 (2012). https://doi.org/10.1007/s13398-011-0024-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-011-0024-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation