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JOURNALS // Sibirskii Zhurnal Vychislitel'noi Matematiki // Archive

Sib. Zh. Vychisl. Mat., 2020 Volume 23, Number 1, Pages 39–51 (Mi sjvm731)

$(m, k)$-schemes for stiff systems of ODEs and DAEs

A. I. Levykinab, A. E. Novikovc, E. A. Novikovcd

a Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090 Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia
c Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660041 Russia
d Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, Akademgorodok 50/44, Krasnoyarsk, 660036 Russia

Abstract: This paper deals with the derivation of the optimal form of the Rosenbrock-type methods in terms of the number of non-zero parameters and computational costs per step. A technique of obtaining $(m, k)$-methods from the well-known Rosenbrock-type methods is justified. There are given formulas for the $(m, k)$-schemes parameters transformation for their two canonical representations and obtaining the form of a stability function. The authors have developed $L$-stable $(3, 2)$-method of order $3$ which requires two evaluations of a function: one evaluation of the Jacobian matrix and one $LU$-decomposition per step. Moreover, in this paper there is formulated an integration algorithm of the alternating step size based on $(3, 2)$-method. It provides the numerical solution for both explicit and implicit systems of ODEs. The numerical results confirming the efficiency of the new algorithm are given.

Key words: Rosenbrock-type methods, differential-algebraic equations, stiff systems of ODEs.

UDC: 519.622

Received: 14.01.2019
Revised: 04.04.2019
Accepted: 15.10.2019

DOI: 10.15372/SJNM20200103


 English version:
Numerical Analysis and Applications, 2020, 13:1, 34–44

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