Geodesic
Two-stage complex Rosenbrock schemes for stiff systems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 270-287 
A. B. Alshin; E. A. Alshina; A. G. Limonov. Two-stage complex Rosenbrock schemes for stiff systems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 270-287. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a5/
@article{ZVMMF_2009_49_2_a5,
     author = {A. B. Alshin and E. A. Alshina and A. G. Limonov},
     title = {Two-stage complex {Rosenbrock} schemes for stiff systems},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {270--287},
     year = {2009},
     volume = {49},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a5/}
}
TY  - JOUR
AU  - A. B. Alshin
AU  - E. A. Alshina
AU  - A. G. Limonov
TI  - Two-stage complex Rosenbrock schemes for stiff systems
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2009
SP  - 270
EP  - 287
VL  - 49
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a5/
LA  - ru
ID  - ZVMMF_2009_49_2_a5
ER  - 
%0 Journal Article
%A A. B. Alshin
%A E. A. Alshina
%A A. G. Limonov
%T Two-stage complex Rosenbrock schemes for stiff systems
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2009
%P 270-287
%V 49
%N 2
%U http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a5/
%G ru
%F ZVMMF_2009_49_2_a5

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

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of $L1$-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single $L2$-stable scheme is found in this family. The coefficients of the fourth-order accurate $L4$-stable scheme previously obtained by P. D. Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of $L$-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.

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

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

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

[4] Marchuk G. I., Shaidurov B. B., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979 | MR

[5] Kalitkin H. H., Alshin A. B., Alshina E. A., Rogov B. V., Vychisleniya na kvaziravnomernykh setkakh, Fizmatlit, M., 2005

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