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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{SJVM_2020_23_1_a2,
     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},
     pages = {39--51},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2020_23_1_a2/}
}
TY  - JOUR
AU  - A. I. Levykin
AU  - A. E. Novikov
AU  - E. A. Novikov
TI  - $(m, k)$-schemes for stiff systems of ODEs and DAEs
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2020
SP  - 39
EP  - 51
VL  - 23
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2020_23_1_a2/
LA  - ru
ID  - SJVM_2020_23_1_a2
ER  - 
%0 Journal Article
%A A. I. Levykin
%A A. E. Novikov
%A E. A. Novikov
%T $(m, k)$-schemes for stiff systems of ODEs and DAEs
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2020
%P 39-51
%V 23
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2020_23_1_a2/
%G ru
%F SJVM_2020_23_1_a2
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/

[1] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and DifferentialAlgebraic Problems, Springer-Verlag, Berlin, 1996 | MR

[2] E. A. Novikov, Yu. V. Shornikov, Komp'yuternoe modelirovanie zhestkih gibridnyh sistem, Izd-vo NGTU, Novosibirsk, 2012

[3] H. H. Rosenbrock, “Some general implicit processes for the numerical solution of differential equations”, Computer, 5 (1963), 329–330 | MR | Zbl

[4] S. S. Artem'ev, G. V. Demidov, “Algoritm peremennogo poryadka i shaga dlya chislennogo resheniya zhestkih sistem obyknovennyh differencial'nyh uravnenii”, Dokl. AN SSSR, 238:3 (1978), 517–520 | MR | Zbl

[5] P. Kaps, G. Wanner, “A study of Rosenbrock-type methods of high order”, Numerische Mathematik, 38 (1981), 279–298 | DOI | MR | Zbl

[6] T. Steihaug, A. Wolfbrandt, “An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations”, Mathematics of Computations, 33 (1979), 521–534 | DOI | MR | Zbl

[7] J. G. Werwer, S. Scholz, J. G. Blom, M. Louter-Nool, “A class of Runge-Kutta-Rosenbrock methods for solving stiff differential equations”, ZAAM, 63 (1983), 13–20 | MR

[8] V. V. Voevodin, Yu. A. Kuznecov, Matricy i vychisleniya, Nauka, Novosibirsk, 1984 | MR

[9] S. P. Norsett, “Restricted Pad`e approximations to the exponential function”, SIAM J. Numer. Analysis, 15:5 (1978), 1008–1029 | DOI | MR | Zbl

[10] A. I. Levykin, Novikov E. A., “A study of $(m, k)$-methods for solving differential-algebraic systems of index $1$”, Mathematical Modeling of Technological Processes, Proc. 8th Int. Conf. CITech 2015, Communication on Computer and Information Science, 549, 2015, 94–107 | DOI

[11] M. Roche, “Rosenbrock methods for differential algebraic equations”, Numerische Mathematik, 52 (1988), 45–63 | DOI | MR | Zbl

[12] A. I. Levykin, A. E. Novikov, E. A. Novikov, “Third order $(m, k)$-method for solving stiff systems of ODEs and DAEs”, Proc. XIV Int. Scientific-Technical Conf. APEIE-2018, Part 4, v. 1, NSTU, Novosibirsk, 2018, 158–163 | MR