Symmetric Runge-Kutta Methods with Higher Derivatives and Quadratic Extrapolation

  • Gennady Yu. Kulikov
  • Ekaterina Yu. Khrustaleva
  • Arkadi I. Merkulov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3991)


In this paper we study the symmetry of Runge-Kutta methods with higher derivatives. We find conditions which provide this property for the above numerical methods. We prove that the family of E-methods constructed earlier consists of symmetric methods only, which lead to the quadratic extrapolation technique in practice.


Adjoint Method Symmetric Method Underlying Method Hermite Type Increment Function 


  1. 1.
    Butcher, J.C.: Numerical methods for ordinary differential equations. John Wiley and Son, Chichester (2003)CrossRefMATHGoogle Scholar
  2. 2.
    Fehlberg, E.: Eine methode zur fehlerverkleinerung bein Runge-Kutta verfahren. ZAMM 38, 421–426 (1958)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Fehlberg, E.: New high-order Runge-Kutta formulas with step size control for systems of first and second order differential equations. ZAMM 44, T17–T19 (1964)MathSciNetGoogle Scholar
  4. 4.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations I: Nonstiff problems. Springer, Berlin (1993)MATHGoogle Scholar
  5. 5.
    Hairer, E., Wanner, G.: Solving ordinary differential equations II: Stiff and differential-algebraic problems. Springer, Berlin (1996)MATHGoogle Scholar
  6. 6.
    Hairer, E., Wanner, G., Lubich, C.: Geometric numerical integration: structure preserving algorithms for ordinary differential equations. Springer, Berlin (2002)MATHGoogle Scholar
  7. 7.
    Kastlunger, K.H., Wanner, G.: Runge-Kutta processes with multiple nodes. Computing 9, 9–24 (1972)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Kastlunger, K.H., Wanner, G.: On Turan type implicit Runge-Kutta methods. Computing 9, 317–325 (1972)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Kulikov, G.Y.: Revision of the theory of symmetric one-step methods for ordinary differential equations. Korean J. Comput. Appl. Math. 5(3), 289–318 (1998)MathSciNetGoogle Scholar
  10. 10.
    Kulikov, G.Y.: On implicit extrapolation methods for ordinary differential equations. Russian J. Numer. Anal. Math. Modelling. 17(1), 41–69 (2002)MathSciNetGoogle Scholar
  11. 11.
    Kulikov, G.Y., Merkulov, A.I., Khrustaleva, E.Y.: On a family of A-stable collocation methods with high derivatives. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2004. LNCS, vol. 3037, pp. 73–80. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Kulikov, G.Y., Merkulov, A.I.: On one-step collocation methods with higher derivatives for solving ordinary differential equations. Zh. Vychisl. Mat. Mat. Fiz. 44(10), 1782–1807 (2004) (in Russian); translation in Comput. Math. Math. Phys. 44(10), 1696–1720 (2004)MathSciNetMATHGoogle Scholar
  13. 13.
    Nørsett, S.P.: One-step methods of Hermite type for numerical integration of stiff systems. BIT 14, 63–77 (1974)CrossRefGoogle Scholar
  14. 14.
    Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Springer, Berlin (1973)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gennady Yu. Kulikov
    • 1
  • Ekaterina Yu. Khrustaleva
    • 2
  • Arkadi I. Merkulov
    • 2
  1. 1.School of Computational and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.Ulyanovsk State UniversityUlyanovskRussia

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