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

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.

1. 1.

   We use this kind of optimization mostly for simplicity reasons. Of course, in general case this condition does not imply \(|R(z)|<1\ \forall z\in {\varOmega }\), so special care should be taken here.

2. 2.

   http://www.unige.ch/~hairer/prog/nonstiff/cprog.tar.

3. 3.

   http://www.unige.ch/~hairer/testset/stiff/hires/res_exact_pic.

Authors and Affiliations

1. Belarusian State University, 220030, Minsk, Belarus

   Boris Faleichik & Ivan Bondar

Corresponding author

Correspondence to Boris Faleichik .

Editors and Affiliations

1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Sofia, Bulgaria

   Ivan Dimov

2. Institue of Mathematics and MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary

   István Faragó

3. University of Rousse, Rousse, Bulgaria

   Lubin Vulkov

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