Geodesic
Investigation of an economical finite difference scheme for an unsteady viscous weakly compressible gas flow
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 4, pp. 701-717 
K. A. Zhukov; A. V. Popov. Investigation of an economical finite difference scheme for an unsteady viscous weakly compressible gas flow. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 4, pp. 701-717. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_4_a12/
@article{ZVMMF_2005_45_4_a12,
     author = {K. A. Zhukov and A. V. Popov},
     title = {Investigation of an economical finite difference scheme for an unsteady viscous weakly compressible gas flow},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {701--717},
     year = {2005},
     volume = {45},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_4_a12/}
}
TY  - JOUR
AU  - K. A. Zhukov
AU  - A. V. Popov
TI  - Investigation of an economical finite difference scheme for an unsteady viscous weakly compressible gas flow
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2005
SP  - 701
EP  - 717
VL  - 45
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_4_a12/
LA  - ru
ID  - ZVMMF_2005_45_4_a12
ER  - 
%0 Journal Article
%A K. A. Zhukov
%A A. V. Popov
%T Investigation of an economical finite difference scheme for an unsteady viscous weakly compressible gas flow
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2005
%P 701-717
%V 45
%N 4
%U http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_4_a12/
%G ru
%F ZVMMF_2005_45_4_a12

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

An implicit finite difference scheme with a splitting operator is constructed for a linear system of equations describing an unsteady viscous weakly compressible gas flow in the case of two spatial variables. The use of a splitting operator made it possible to propose an algorithm for seeking a difference solution in explicit form by the Fourier method. For the numerical solution obtained with this scheme, an error estimate is proved depending on the parameter characterizing the gas compressibility and gas viscosity. The numerical results presented show the efficiency of this method in comparison with an implicit scheme based on iterative conjugate gradient methods.

[1] Popov A. V., “Raznostnaya skhema dlya nestatsionarnogo dvizheniya vyazkogo slaboszhimaemogo gaza”, Optimizatsiya chisl. metodov, IM VTs UNTs RAN, UFA, 2000, 115–160

[2] Popov A. V., Of finite difference sheme for viscous weakly compressible gas problem, Rept No 9617, Dept. Math. Univ. Nijmengen, The Netherlands, 1996

[3] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Nauka, M., 1970 | MR

[4] Temam R., Uravneniya Nave-Stoksa, Mir, M., 1981 | MR | Zbl

[5] Yanenko N. N., Metod drobnykh shagov resheniya mnogomernykh zadach matematicheskoi fiziki, Novosibirskii gos. un-t, Novosibirsk, 1966

[6] Prohl A., Projection and qvuasi-compressibility methods for solving the imcompressible Navie-Stokes equations, Teubner, Stuttgard, 1997 | MR | Zbl

[7] Kobelkov G. M., Valedinskii V. D., O raznostnom analoge neravenstva $||p||'\leq c||\mathrm{grad}||_{-1}$, Preprint No 67, OVM AN SSSR, M., 1983 | MR

[8] Kobelkov G. M., Valedinskiy V. D., “On the inequality $||p||\leq c||\mathrm{grad}||_{-1}$ and its finite dimensional image”, Soviet J. Numer Analys. and Math. Modelling, 1:3 (1986), 189–201 | DOI | MR

[9] Amosov A. A., Zlotnik A. A., “Raznostnye skhemy vtorogo poryadka tochnosti dlya uravnenii odnomernogo dvizheniya vyazkogo gaza”, Zh. vychisl. matem. i matem. fiz., 27:7 (1987), 1032–1049 | MR | Zbl

[10] Sobolev S. L., Vaskevich V. L., Kvadraturnye formuly, IM SO RAN, Novosibirsk, 1996 | Zbl

[11] Zhukov K. A., “Raznostnaya skhema dlya nestatsionarnykh techenii vyazkogo slaboszhimamogo gaza”, Chisl. metody resheniya lineinykh i nelineinykh kraevykh zadach, Materialy vseros. molodezhnoi nauchn. shkoly-konferentsii, Tr. Matem. tsentra im. N. I. Lobachevskogo, 13, Kazanskoe matem. ob-vo, Kazan, 2001, 178–184 | MR

[12] Chizhonkov E. V., Relaksatsionnye metody resheniya sedlovykh zadach, IVM RAN, M., 2002

[13] Zhukov K. A., Popov A. V., “Ekonomichnaya raznostnaya skhema dlya nestatsionarnogo dvizheniya vyazkogo slaboszhimaemogo gaza”, Chisl. metody resheniya lineinykh i nelineinykh kraevykh zadach, Materialy II vseros. molodezhnoi nauchn. shkoly-konferentsii, Tr. Matem. tsentra im. N. I. Lobachevskogo, Kazan, 2003, 119–128 | MR

[14] Kobelkov G. M., “O chislennykh metodakh resheniya uravnenii Nave–Stoksa v peremennykh skorost-davlenie”, Vychisl. protsessy i sistemy, 8, Nauka, M., 1991, 204–236 | MR