Geodesic
How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1126-1141 
L. M. Skvortsov. How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1126-1141. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a4/
@article{ZVMMF_2017_57_7_a4,
     author = {L. M. Skvortsov},
     title = {How to avoid accuracy and order reduction in {Runge{\textendash}Kutta} methods as applied to stiff problems},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1126--1141},
     year = {2017},
     volume = {57},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a4/}
}
TY  - JOUR
AU  - L. M. Skvortsov
TI  - How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2017
SP  - 1126
EP  - 1141
VL  - 57
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a4/
LA  - ru
ID  - ZVMMF_2017_57_7_a4
ER  - 
%0 Journal Article
%A L. M. Skvortsov
%T How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2017
%P 1126-1141
%V 57
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a4/
%G ru
%F ZVMMF_2017_57_7_a4

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

The solution of stiff problems is frequently accompanied by a phenomenon known as order reduction. The reduction in the actual order can be avoided by applying methods with a fairly high stage order, ideally coinciding with the classical order. However, the stage order sometimes fails to be increased; moreover, this is not possible for explicit and diagonally implicit Runge–Kutta methods. An alternative approach is proposed that yields an effect similar to an increase in the stage order. New implicit and stabilized explicit Runge–Kutta methods are constructed that preserve their order when applied to stiff problems.

[1] Alexander R., “Diagonally implicit Runge-Kutta methods for stiff O. D.E.'s”, SIAM J. Numer. Anal., 14:6 (1977), 1006–1021 | DOI | MR | Zbl

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

[3] Skvortsov L. M., “Diagonalno neyavnye FSAL-metody Runge-Kutty dlya zhestkikh i differentsialno-algebraicheskikh sistem”, Matem. modelirovanie, 14:2 (2002), 3–17

[4] Kværmø A., “Singly diagonally implicit Runge–Kutta methods with an explicit first stage”, BIT, 44:3 (2004), 489–502 | DOI | MR | Zbl

[5] Skvortsov L. M., “Diagonalno neyavnye metody Runge-Kutty dlya zhestkikh zadach”, Zh. vychisl. matem. i matem. fiz., 46:12 (2006), 2209–2222

[6] Rang J., “An analysis of the Prothero-Robinson example for constructing new adaptive ESDIRK methods of order 3 and 4”, Appl. Numer. Math., 94 (2015), 75–87 | DOI | MR | Zbl

[7] Prothero A., Robinson A., “On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations”, Math. Comput., 28:1 (1974), 145–162 | DOI | MR | Zbl

[8] Dekker K., Verver Ya., Ustoichivost metodov Runge-Kutty dlya zhestkikh nelineinykh differentsialnykh uravnenii, Mir, M., 1988

[9] Sans-Serna J. M., Verwer J. G., Hundsdorfer W. H., “Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations”, Numer. Math., 59 (1986), 405–418 | DOI | MR

[10] Skvortsov L. M., “Tochnost metodov Runge-Kutty pri reshenii zhestkikh zadach”, Zh. vychisl. matem. i matem. fiz., 43:9 (2003), 1374–1384 | Zbl

[11] Skvortsov L. M., “Modelnye uravneniya dlya issledovaniya tochnosti metodov Runge-Kutty”, Matem. modelirovanie, 22:5 (2010), 146–160

[12] Skvortsov L. M., “Yavnye metody Runge-Kutty dlya umerenno zhestkikh zadach”, Zh. vychisl. matem. i matem. fiz., 45:11 (2005), 2017–2030 | Zbl

[13] Butcher J. C., Chen D. J. L., “A new type of singly implicit Runge-Kutta methods”, Appl. Numer. Math., 34:2–3 (2000), 179–188 | DOI | MR | Zbl

[14] Skvortsov L. M., “Odnokratno neyavnye diagonalno rasshirennye metody Runge-Kutty chetvertogo poryadka”, Zh. vychisl. matem. i matem. fiz., 54:5 (2014), 755–765 | DOI | Zbl

[15] Gantmakher F. R., Teoriya matrits, 4-e izd., Nauka, M., 1988

[16] Skvortsov L. M., “Ekonomichnaya skhema realizatsii neyavnykh metodov Runge-Kutty”, Zh. vychisl. matem. i matem. fiz., 48:11 (2008), 2008–2018

[17] Skvortsov L. M., “Diagonalno-neyavnye metody Runge-Kutty dlya differentsialno-algebraicheskikh uravnenii indeksov 2 i 3”, Zh. vychisl. matem. i matem. fiz., 50:6 (2010), 1047–1059 | Zbl

[18] Lebedev V. I., “Kak reshat yavnymi metodami zhestkie sistemy differentsialnykh uravnenii”, Vychislitelnye protsessy i sistemy, 8, Nauka, M., 1991, 237–291 | Zbl

[19] Skvortsov L. M., “Yavnye stabilizirovannye metody Runge-Kutty”, Zh. vychisl. matem. i matem. fiz., 51:7 (2011), 1236–1250 | Zbl

[20] Khairer E., Nersett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990

[21] Kalitkin N. N., Poshivailo I. P., “Vychisleniya s ispolzovaniem obratnykh skhem Runge-Kutty”, Matem. modelirovanie, 25:10 (2013), 79–96

[22] Cash J. R., Singhal A., “Mono-implicit Runge-Kutta formulae for the numerical integration of stiff differential systems”, IMA J. Numer. Anal., 2 (1982), 211–227 | DOI | MR | Zbl

[23] Burrage K., Chipman F. H., Muir P. H., “Order results for mono-implicit Runge-Kutta methods”, SIAM J. Numer. Anal., 31:3 (1994), 876–891 | DOI | MR | Zbl

[24] Kulikov G. Yu., Shindin S. K., “Adaptive nested implicit Runge-Kutta formulas of Gauss type”, Appl. Numer. Math., 59:3–4 (2009), 707–722 | DOI | MR | Zbl

[25] Kulikov G. Yu., “Vlozhennye simmetrichnye neyavnye gnezdovye metody Runge-Kutty tipov Gaussa i Lobatto dlya resheniya zhestkikh obyknovennykh differentsialnykh uravnenii i gamiltonovykh sistem”, Zh. vychisl. matem. i matem. fiz., 55:6 (2015), 986–1007 | DOI | Zbl

[26] Bogacki P., Shampine L. F., “A 3(2) pair of Runge-Kutta formulas”, Appl. Math. Lett., 2:4 (1989), 321–325 | DOI | MR | Zbl