BIT Numerical Mathematics

, Volume 51, Issue 3, pp 529–553 | Cite as

B-series analysis of iterated Taylor methods

  • Kristian Debrabant
  • Anne Kværnø
Open Access


For stochastic implicit Taylor methods that use an iterative scheme to compute their numerical solution, stochastic B-series and corresponding growth functions are constructed. From these, convergence results based on the order of the underlying Taylor method, the choice of the iteration method, the predictor, and the number of iterations, for Itô and Stratonovich SDEs, and for weak as well as strong convergence are derived. As special case, also the application of Taylor methods to ODEs is considered. The theory is supported by numerical experiments.


Stochastic Taylor method Stochastic differential equation Iterative scheme Order Newton’s method Weak approximation Strong approximation Growth function Stochastic B-series 

Mathematics Subject Classification (2000)

65C30 60H35 65C20 68U20 


  1. 1.
    Barrio, R.: Performance of the Taylor series method for ODEs/DAEs. Appl. Math. Comput. 163(2), 525–545 (2005). doi: 10.1016/j.amc.2004.02.015 MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Burrage, K., Burrage, P.M.: High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math. 22(1–3), 81–101 (1996). doi: 10.1016/S0168-9274(96)00027-X. Special issue celebrating the centenary of Runge–Kutta methods MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Burrage, K., Burrage, P.M.: Order conditions of stochastic Runge–Kutta methods by B-series. SIAM J. Numer. Anal. 38(5), 1626–1646 (2000). doi: 10.1137/S0036142999363206 (electronic) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Burrage, K., Tian, T.: Implicit stochastic Runge–Kutta methods for stochastic differential equations. BIT 44(1), 21–39 (2004). doi: 10.1023/B:BITN.0000025089.50729.0f MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Burrage, P.M.: Runge–Kutta methods for stochastic differential equations. Ph.D. thesis, The University of Queensland, Brisbane (1999) Google Scholar
  6. 6.
    Butcher, J.: An algebraic theory of integration methods. Math. Comput. 26, 79–106 (1972). doi: 10.2307/2004720 MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Debrabant, K., Kværnø, A.: B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47(1), 181–203 (2008/2009). doi: 10.1137/070704307 CrossRefGoogle Scholar
  8. 8.
    Debrabant, K., Kværnø, A.: Composition of stochastic B-series with applications to implicit Taylor methods. Appl. Numer. Math. 61(4), 501–511 (2011). doi: 10.1016/j.apnum.2010.11.014 MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Debrabant, K., Kværnø, A.: Stochastic Taylor expansions: weight functions of B-series expressed as multiple integrals. Stoch. Anal. Appl. 28(2), 293–302 (2010). doi: 10.1080/07362990903546504 MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006) MATHGoogle Scholar
  11. 11.
    Hairer, E., Wanner, G.: On the Butcher group and general multi-value methods. Computing 13(1), 1–15 (1974) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000). doi: 10.1137/S003614299834736X (electronic) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Jackson, K.R., Kværnø, A., Nørsett, S.P.: The use of Butcher series in the analysis of Newton-like iterations in Runge–Kutta formulas. Appl. Numer. Math. 15(3), 341–356 (1994). doi: 10.1016/0168-9274(94)00031-X. International Conference on Scientific Computation and Differential Equations (Auckland, 1993) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jackson, K.R., Kværnø, A., Nørsett, S.P.: An analysis of the order of Runge–Kutta methods that use an iterative scheme to compute their internal stage values. BIT 36(4), 713–765 (1996). doi: 10.1007/BF01733789 MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics, vol. 21, 2nd edn. Springer, Berlin (1999) Google Scholar
  16. 16.
    Rößler, A.: Rooted tree analysis for order conditions of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations. Stoch. Anal. Appl. 24(1), 97–134 (2006). doi: 10.1080/07362990500397699 MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Tian, T., Burrage, K.: Implicit Taylor methods for stiff stochastic differential equations. Appl. Numer. Math. 38(1–2), 167–185 (2001). doi: 10.1016/S0168-9274(01)00034-4 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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