ABSTRACT
A form of Rosenbrock-type methods optimal in terms of the number of non-zero parameters and computational costs per step is considered. A technique of obtaining \((m, k)\)-methods from some well-known Rosenbrock-type methods is justified. Formulas for transforming the parameters of \((m,k)\)-schemes and for obtaining a stability function are given for two canonical representations of the schemes. An \(L\)-stable \((3, 2)\)-method of order 3 is proposed, which requires two evaluations of the function: one evaluation of the Jacobian matrix and one \(LU\)-decomposition per step. A variable step size integration algorithm based on the \((3,2)\)-method is formulated. It provides a numerical solution for both explicit and implicit systems of ODEs. Numerical results are presented to show the efficiency of the new algorithm.
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Funding
This work was supported by the Russian Foundation for Basic Research (project no. 17-07-01513 A). The work of the first author was supported by ICM&MG SB RAS (state assignment no. 0315-2019-0002).
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Levykin, A.I., Novikov, A.E. & Novikov, E.A. Schemes of (m, k)-Type for Solving Differential-Algebraic and Stiff Systems. Numer. Analys. Appl. 13, 34–44 (2020). https://doi.org/10.1134/S1995423920010036
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DOI: https://doi.org/10.1134/S1995423920010036