(m, k)-schemes for stiff systems of ODEs and DAEs
A.I. Levykin1,2, A.E. Novikov3, E.A. Novikov3,4
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
3Siberian Federal University, Krasnoyarsk, Russia
4Institute of Computational Mathematicsof the Federal Research Centerof the Krasnoyarsk Scientific Center,Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia
Keywords: методы типа Розенброка, дифференциально-алгебраические уравнения, жесткие системы ОДУ, Rosenbrock-type methods, differential-algebraic equations, stiff systems of ODEs
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.
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