Abstract:
The solution of stiff problems is frequently accompanied by a phenomenon known as order reduction. The reduction in the actual order can be avoided by applying methods with a fairly high stage order, ideally coinciding with the classical order. However, the stage order sometimes fails to be increased; moreover, this is not possible for explicit and diagonally implicit Runge–Kutta methods. An alternative approach is proposed that yields an effect similar to an increase in the stage order. New implicit and stabilized explicit Runge–Kutta methods are constructed that preserve their order when applied to stiff problems.
Key words:
Runge–Kutta methods, stiff problems, order reduction.
Citation:
L. M. Skvortsov, “How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems”, Zh. Vychisl. Mat. Mat. Fiz., 57:7 (2017), 1126–1141; Comput. Math. Math. Phys., 57:7 (2017), 1124–1139
\Bibitem{Skv17}
\by L.~M.~Skvortsov
\paper How to avoid accuracy and order reduction in Runge--Kutta methods as applied to stiff problems
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2017
\vol 57
\issue 7
\pages 1126--1141
\mathnet{http://mi.mathnet.ru/zvmmf10586}
\crossref{https://doi.org/10.7868/S0044466917070134}
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\transl
\jour Comput. Math. Math. Phys.
\yr 2017
\vol 57
\issue 7
\pages 1124--1139
\crossref{https://doi.org/10.1134/S0965542517070119}
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Linking options:
https://www.mathnet.ru/eng/zvmmf10586
https://www.mathnet.ru/eng/zvmmf/v57/i7/p1126
This publication is cited in the following 7 articles:
Hao Wang, Chuanda Wang, Gang Wang, Yongjun Pan, Aki Mikkola, Haijun Peng, “A symplectic finite element method in time for periodic response of multibody systems”, Multibody Syst Dyn, 2025
Abhijit Biswas, David Ketcheson, Benjamin Seibold, David Shirokoff, “Algebraic Structure of the Weak Stage Order Conditions for Runge–Kutta Methods”, SIAM J. Numer. Anal., 62:1 (2024), 48
Gennady Yu. Kulikov, Maria V. Kulikova, Studies in Systems, Decision and Control, 539, State Estimation for Nonlinear Continuous–Discrete Stochastic Systems, 2024, 111
L. M. Skvortsov, “Generalizations of the Stage Order of Runge–Kutta Methods”, Comput. Math. and Math. Phys., 64:12 (2024), 2796
L. M. Skvortsov, “Third- and fourth-order ESDIRK methods for stiff and differential-algebraic problems”, Comput. Math. Math. Phys., 46:5 (2022), 766–783
G. Yu. Kulikov, “Nested implicit Runge–Kutta pairs of Gauss and Lobatto types with local and global error controls for stiff ordinary differential equations”, Comput. Math. Math. Phys., 60:7 (2020), 1134–1154
L. M. Skvortsov, “Implicit Runge–Kutta methods with explicit internal stages”, Comput. Math. Math. Phys., 58:3 (2018), 307–321
Citation in format AMSBIB
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