Geodesic
Three- and four-step implicit absolutely stable fourth-order Runge–Kutta schemes
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 116-130 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two types of implicit fourth-order Runge–Kutta schemes are constructed for first-order ordinary differential equations, multidimensional transfer equations, and compressible gas equations. The absolute stability of the schemes is proved by applying the principle of frozen coefficients. Adaptive artificial viscosity ensuring good time convergence and oscillations damping near discontinuities is used in solving gas dynamics equations. The comparative efficiency of the schemes is illustrated by numerical results obtained for compressible gas flows. 
@article{ZVMMF_2006_46_1_a11,
     author = {V. I. Pinchukov},
     title = {Three- and four-step implicit absolutely stable fourth-order {Runge{\textendash}Kutta} schemes},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {116--130},
     year = {2006},
     volume = {46},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a11/}
}
TY  - JOUR
AU  - V. I. Pinchukov
TI  - Three- and four-step implicit absolutely stable fourth-order Runge–Kutta schemes
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 116
EP  - 130
VL  - 46
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a11/
LA  - ru
ID  - ZVMMF_2006_46_1_a11
ER  - 
%0 Journal Article
%A V. I. Pinchukov
%T Three- and four-step implicit absolutely stable fourth-order Runge–Kutta schemes
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 116-130
%V 46
%N 1
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a11/
%G ru
%F ZVMMF_2006_46_1_a11
V. I. Pinchukov. Three- and four-step implicit absolutely stable fourth-order Runge–Kutta schemes. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 1, pp. 116-130. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_1_a11/

[1] Rusanov V. V., “Raznostnye skhemy tretego poryadka tochnosti dlya skvoznogo rascheta razryvnykh reshenii”, Dokl. AN SSSR, 180:6 (1968), 1303–1305 | MR | Zbl

[2] Burstein S. Z., Mirin A. A., “Third order difference methods for hyperbolic equations”, J. Comput. Phys., 5:3 (1970), 547–571 | DOI | MR | Zbl

[3] Kutler P., Lomax H., Warming R., Computation of space shuttle flow fields using noncentered finite-difference schemes, AIAA Paper No 72-193, 1972, 25 pp.

[4] Shu C.-W., “Total-variation-diminishing time discretizations”, SIAM J. Sci. Statist. Comput., 9:6 (1988), 1073–1084 | DOI | MR | Zbl

[5] Osher S., Shu C.-W., “Efficient implementation of essentially nonoscillatory shock-capturring schemes”, J. Comput. Phys., 77:2 (1988), 439–471 | DOI | MR | Zbl

[6] Pinchukov V. I., Shu Ch.-V., Chislennye metody vysokikh poryadkov dlya zadach aerogidrodinamiki, Izd-vo SO RAN, Novosibirsk, 2000

[7] Tolstykh A. I., Kompaktnye raznostnye skhemy i ikh prilozheniya k problemam aerogidrodinamiki, Nauka, M., 1990 | MR

[8] Pinchukov V. I., “Absolyutno ustoichivye skhemy Runge–Kutty tretego poryadka approksimatsii”, Zh. vychisl. matem. i matem. fiz., 39:11 (1999), 1855–1868 | MR | Zbl

[9] Pinchukov V. I., “Sravnenie neyavnykh skhem Runge–Kutty tretego poryadka”, Vychisl. tekhnologii, 7:5 (2002), 44–57 | MR | Zbl

[10] Pinchukov V. I., “O neyavnykh absolyutno ustoichivykh skhemakh Runge–Kutty chetvertogo poryadka”, Vychisl. tekhnologii, 7:1 (2002), 96–105 | MR | Zbl

[11] Pinchukov V. I., “O chislennom issledovanii transzvukovykh turbulentnykh techenii vozle kryla neyavnymi skhemami vysokikh poryadkov”, Vychisl. tekhnologii, 6:2 (2001), 110–121 | MR | Zbl

[12] Pinchukov V. I., “Nelineinye setochnye filtry i ikh ispolzovanie v skhemakh vysokikh poryadkov dlya zadach aerodinamiki”, Matem. modelirovanie, 10:11 (1999), 111–115 | MR

[13] Jameson A., Schmidt W., Turcel E., Numerical solution of the Euler equations by finite-volume method using Runge–Kutta time stepping schemes, AIAA Paper 81-1259, 1981 | Zbl