Geodesic
$(m, k)$-schemes for stiff systems of ODEs and DAEs
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 23 (2020) no. 1, pp. 39-51

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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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     author = {A. I. Levykin and A. E. Novikov and E. A. Novikov},
     title = {$(m, k)$-schemes for stiff systems of {ODEs} and {DAEs}},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2020_23_1_a2/}
}
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A. I. Levykin; A. E. Novikov; E. A. Novikov. $(m, k)$-schemes for stiff systems of ODEs and DAEs. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 23 (2020) no. 1, pp. 39-51. http://geodesic.mathdoc.fr/item/SJVM_2020_23_1_a2/