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Stability Properties of Repeated Richardson Extrapolation Applied Together with Some Implicit Runge-Kutta Methods

  • Zahari Zlatev
  • Ivan Dimov
  • István Faragó
  • Krassimir Georgiev
  • Ágnes HavasiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

Repeated Richardson Extrapolation can successfully be used in the efforts to improve the efficiency of the numerical treatment of systems of ordinary differential equations (ODEs) mainly by increasing the accuracy of the computed results. It is assumed in this paper that Implicit Runge-Kutta Methods (IRKMs) are used in the numerical solution of systems of ODEs. If the order of accuracy of the selected IRKM is p, then the order of accuracy of its combination with the Repeated Richardson Extrapolation is at least \(p+2\) (assuming here that the right-hand-side of the system of ODEs is sufficiently many times continuously differentiable). However, it is additionally necessary to establish that the absolute stability properties of the new numerical methods (that are combinations of the Repeated Richardson Extrapolation and the selected IRKMs) are preserved, and this is an extremely difficult problem. Results related to the stability of the computations are derived and numerical tests with a two-parameter system of three ODEs and an atmospheric chemical scheme with 56 compounds, which is defined mathematically by a very stiff and ill-conditioned system of non-linear ODEs, are presented. The research results described in this paper can be considered as a continuation of the study carried out in Zlatev et al.: Richardson Extrapolation: Practical Aspects and Applications. De Gruyter, Berlin (2017).

Keywords

Systems of ordinary differential equations (ODEs) Implicit Runge-Kutta Methods Repeated Richardson Extrapolation Absolute stability properties Atmospheric chemical schemes 

Notes

Acknowledgement

The work is partially supported by the Bulgarian National Science Fund under grant DFNI 12/5 and by the Hungarian Research Fund OTKA under grant no. K-112157 and SNN-125119.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Environmental ScienceAarhus UniversityRoskildeDenmark
  2. 2.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria
  3. 3.MTA-ELTE Numerical Analysis and Large Networks Research GroupBudapestHungary
  4. 4.Department of Applied Analysis and Computational MathematicsEötvös Loránd UniversityBudapestHungary

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