Matrix-Free Iterative Processes for Implementation of Implicit Runge–Kutta Methods

  • Boris FaleichikEmail author
  • Ivan Bondar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


In this work we present so-called generalized Picard iterations (GPI) – a family of iterative processes which allows to solve mildly stiff ODE systems using implicit Runge–Kutta (IRK) methods without storing and inverting Jacobi matrices. The key idea is to solve nonlinear equations arising from the base IRK method by special iterative process based on the idea of artificial time integration. By construction these processes converge for all asymptotically stable linear ODE systems and all A-stable base IRK methods at arbitrary large time steps. The convergence rate is limited by the value of “stiffness ratio”, but not by the value of Lipschitz constant of Jacobian. The computational scheme is well suited for parallelization on systems with shared memory. The presented numerical results exhibit that the proposed GPI methods in case of mildly stiff problems can be more advantageous than traditional explicit RK methods.


Runge–Kutta methods Stiff problems Parallel methods 


  1. 1.
    Faleichik, B., Bondar, I., Byl, V.: Generalized Picard iterations: a class of iterated Runge-Kutta methods for stiff problems. J. Comp. Appl. Math. 262, 37–50 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Faleichik, B.V.: Explicit implementation of collocation methods for stiff systems with complex spectrum. J. Numer. Anal. 5(1–2), 49–59 (2010)MathSciNetGoogle Scholar
  3. 3.
    Hairer, E., Wanner, G.: Solving ordinary differential equations II. In: Hairer, E., Wanner, G. (eds.) Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Heidelberg (1996)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Belarusian State UniversityMinskBelarus

Personalised recommendations