Geodesic
Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the ODE systems
Matematičeskoe modelirovanie, Tome 23 (2011) no. 1, pp. 87-99.

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

Two schemes of the third and fourth order are presented on the basis of the schemes including the right parts derivatives. The iterative procedure is suggested for the realization of these schemes combined with using of the complex-valued arithmetic. Sufficiency of only two Newtonian iterations is shown for the one time step calculation. The procedure for local error estimation based on results of two sequential iterations has been proposed and approved.
Keywords: stiff differential equations systems, implicit schemes with derivatives, complex-valued realization. 
@article{MM_2011_23_1_a7,
     author = {Yu. A. Sigunov and I. R. Didenko},
     title = {Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the {ODE} systems},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {87--99},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2011_23_1_a7/}
}
TY  - JOUR
AU  - Yu. A. Sigunov
AU  - I. R. Didenko
TI  - Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the ODE systems
JO  - Matematičeskoe modelirovanie
PY  - 2011
SP  - 87
EP  - 99
VL  - 23
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2011_23_1_a7/
LA  - ru
ID  - MM_2011_23_1_a7
ER  - 
%0 Journal Article
%A Yu. A. Sigunov
%A I. R. Didenko
%T Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the ODE systems
%J Matematičeskoe modelirovanie
%D 2011
%P 87-99
%V 23
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2011_23_1_a7/
%G ru
%F MM_2011_23_1_a7
Yu. A. Sigunov; I. R. Didenko. Complex-valued realization of the implicit single-stage methods up to fourth order for numerical integration of the ODE systems. Matematičeskoe modelirovanie, Tome 23 (2011) no. 1, pp. 87-99. http://geodesic.mathdoc.fr/item/MM_2011_23_1_a7/

[1] Oran E., Boris Dzh., Chislennoe modelirovanie reagiruyuschikh potokov, Mir, M., 1990, 660 pp. | MR

[2] Kholodniok M., Klich A., Kubichek M., Marek M., Metody analiza nelineinykh dinamicheskikh modelei, Mir, M., 1991, 368 pp.

[3] Khairer E., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Zhestkie i differentsialno-algebraicheskie zadachi, Mir, M., 1999, 685 pp.

[4] Kalitkin N. N., “Chislennye metody resheniya zhestkikh sistem”, Matem. modelirovanie, 7:5 (1995), 8–11 | Zbl

[5] Rosenbrock H. H., “Some general implicit processes for the numerical solution of differential equations”, Comput. J., 5:4 (1963), 329–330 | DOI | MR | Zbl

[6] Kalitkin N. N., Ritus I. V., Kompleksnaya skhema resheniya parabolicheskikh uravnenii, prepr. No 32, IPM im. M. V. Keldysha, M., 1981, 31 pp. | MR

[7] Dnestrovskaya E. Yu., Kalitkin N. N., Skhemy Rozenbroka s kompleksnymi koeffitsientami dlya uravneniya perenosa, prepr. No 66, IPM im. M. V. Keldysha, M., 1990, 22 pp. | MR

[8] Dnestrovskaya E. Yu., Kalitkin N. N., Ritus I. V., “Reshenie uravnenii v chastnykh proizvodnykh skhemami s kompleksnymi koeffitsientami”, Matem. modelirovanie, 3:9 (1991), 114–127 | MR | Zbl

[9] Alshin A. B., Alshina E. A., Kalitkin N. N., Koryagina A. B., “Skhemy Rozenbroka s kompleksnymi koeffitsientami dlya zhestkikh i differentsialno-algebraicheskikh sistem”, Zh. vychisl. matem. i matem. fiz., 46:8 (2006), 1392–1414 | MR

[10] Dekker K., Verver Ya., Ustoichivost metodov Runge-Kutty dlya zhestkikh nelineinykh differentsialnykh uravnenii, Mir, M., 1988, 334 pp. | MR

[11] Medvedskii R. I., Sigunov Yu. A., “O reshenii odnomernykh nelineinykh zadach teploprovodnosti na izotermicheskoi setke”, Zh. vychisl. matem. i matem. fiz., 29:11 (1989), 1742–1746 | MR

[12] Gelinas R. J., Doss S. K., Miller K., “The moving finite element method: application to general partial differential equations with multiple large gradients”, J. Comput. Phys., 40 (1981), 202–249 | DOI | MR | Zbl

[13] Shirkov P. D., “Optimalno zatukhayuschie skhemy s kompleksnymi koeffitsientami dlya zhestkikh sistem ODU”, Matem. modelirovanie, 4:8 (1992), 47–57 | MR | Zbl

[14] Alshin A. B., Alshina E. A., Limonov A. G., “Dvukhstadiinye kompleksnye skhemy Rozenbroka dlya zhestkikh sistem”, Zh. vychisl. matem. i matem. fiz., 49:2 (2009), 270–287 | MR | Zbl