Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations

  • H. De la Cruz
  • R. J. Biscay
  • F. Carbonell
  • J. C. Jimenez
  • T. Ozaki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3991)


A new class of stable methods for solving ordinary differential equations (ODEs) is introduced. This is based on combining the Local Linearization (LL) integrator with other extant discretization methods. For this, an auxiliary ODE is solved to determine a correction term that is added to the LL approximation. In particular, combining the LL method with (explicit) Runge Kutta integrators yields what we call LLRK methods. This permits to improve the order of convergence of the LL method without loss of its stability properties. The performance of the proposed integrators is illustrated through computer simulations.


Phase Portrait Local Linearization Matrix Exponential General Linear Method Local Linearization Approximation 


  1. 1.
    Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-Explicit Runge-Kutta methods for time-depending partial differential equations. Appl. Numer. Math. 25, 151–167 (1995)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Beyn, W.J.: On the numerical approximation of phase portraits near stationary points. SIAM J.Numer. Anal. 24, 1095–1113 (1987)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Biscay, R.J., De la Cruz, H., Carbonell, F., Ozaki, T., Jimenez, J.C.: A Higher Order Local Linearization Method for Solving Ordinary Differential Equations. Technical Report, Instituto de Cibernetica, Matematica y Fisica, La Habana (2005)Google Scholar
  4. 4.
    Bower, J.M., Beeman, D.: The book of GENESIS: exploring realistic neural models with the general neural simulation system. Springer, Heidelberg (1995)MATHGoogle Scholar
  5. 5.
    Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods Chichester. John Wiley & Sons, Chichester (1987)MATHGoogle Scholar
  6. 6.
    Carroll, J.: A matricial exponentially fitted scheme for the numerical solution of stiff initial-value problems. Computers Math. Applic. 26, 57–64 (1993)MATHCrossRefGoogle Scholar
  7. 7.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, 3rd edn. Springer, Berlin (1996)MATHGoogle Scholar
  8. 8.
    Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. Numerical Analysis Report 452, Manchester Centre for Computational Mathematics (2004)Google Scholar
  9. 9.
    Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jimenez, J.C.: A Simple Algebraic Expression to Evaluate the Local Linearization Schemes for Stochastic Differential Equations. Appl. Math. Lett. 15, 775–780 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jimenez, J.C., Biscay, R.J., Mora, C.M., Rodriguez, L.M.: Dynamic properties of the Local Linearization method for initial-valued problems. Appl. Math. Comput. 126, 63–81 (2002)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jimenez, J.C., Carbonell, F.: Rate of convergence of local linearization schemes for initial-value problems. Appl. Math. Comput. 171, 1282–1295 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer 11, 341–434 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Moler, C., Van Loan, C.F.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 3–49 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ozaki, T.: A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach. Statist. Sinica. 2, 113–135 (1992)MATHMathSciNetGoogle Scholar
  16. 16.
    Ramos, J.I., Garcia-Lopez, C.M.: Piecewise-linearized methods for initial-value problems. Appl. Math. Comput. 82(1992), 273–302 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • H. De la Cruz
    • 1
    • 2
  • R. J. Biscay
    • 3
  • F. Carbonell
    • 3
  • J. C. Jimenez
    • 3
  • T. Ozaki
    • 4
  1. 1.Universidad de GranmaBayamoCuba
  2. 2.Universidad de las Ciencias InformáticasLa HabanaCuba
  3. 3.Instituto de CibernéticaMatemática y FísicaLa HabanaCuba
  4. 4.Institute of Statistical MathematicsTokyoJapan

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