Geodesic
Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1304-1316 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For the quasi-gasdynamic system of equations, there holds the law of nondecreasing entropy. Difference methods based on this system have been successfully used in numerous applications and test gasdynamic computations. In theoretical terms, however, for standard spatial discretizations of this system, the nondecreasing entropy law does not hold exactly even in the one-dimensional case because of the mesh imbalance terms. For the quasi-gasdynamic equations, a new conservative spatial discretization is proposed for which the entropy balance equation has an appropriate form and the entropy production is guaranteed to be nonnegative (which also holds in the presence of body forces and heat sources). An important element of this discretization is that it makes use of nonstandard space-averaging techniques, including a nonlinear “logarithmic” averaging of the density and internal energy. The results hold on arbitrary nonuniform meshes. 
@article{ZVMMF_2012_52_7_a12,
     author = {A. A. Zlotnik},
     title = {Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1304--1316},
     year = {2012},
     volume = {52},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a12/}
}
TY  - JOUR
AU  - A. A. Zlotnik
TI  - Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2012
SP  - 1304
EP  - 1316
VL  - 52
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a12/
LA  - ru
ID  - ZVMMF_2012_52_7_a12
ER  - 
%0 Journal Article
%A A. A. Zlotnik
%T Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2012
%P 1304-1316
%V 52
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a12/
%G ru
%F ZVMMF_2012_52_7_a12
A. A. Zlotnik. Spatial discretization of the one-dimensional quasi-gasdynamic system of equations and the entropy balance equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1304-1316. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_7_a12/

[1] Chetverushkin B.N., Kineticheskie skhemy i kvazigazodinamicheskaya sistema uravnenii, MAKS Press, M., 2004

[2] Elizarova T.G., Kvazigazodinamicheskie uravneniya i metody rascheta vyazkikh techenii, Nauchnyi mir, M., 2007

[3] Sheretov Yu.V., Dinamika sploshnykh sred pri prostranstvenno-vremennóm osrednenii, Regulyarnaya i khaoticheskaya dinamika, M.-Izhevsk, 2009

[4] Landau L.D., Lifshits E.M., Teoreticheskaya fizika, v. VI, Gidrodinamika, Izd. 3-e, Nauka, M., 1986

[5] Godunov S.K., Romenskii E.I., Elementy mekhaniki sploshnoi sredy i zakony sokhraneniya, Nauchnaya kniga, Novosibirsk, 1998

[6] Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, ed. S.K. Godunov, Nauka, M., 1976

[7] Prokopov G.P., “Neobkhodimost kontrolya entropii v gazodinamicheskikh raschetakh”, Zh. vychisl. matem. i matem. fiz., 47:9 (2007), 1591–1601 | MR

[8] Tadmor E., “Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems”, Acta Numerica, 12 (2003), 451–512 | DOI | MR | Zbl

[9] Antontsev S.N., Kazhikhov A.V., Monakhov V.N., Kraevye zadachi mekhaniki neodnorodnykh zhidkostei, Nauka, 1983

[10] Amosov A.A., Zlotnik A.A., “A study of finite-difference method for the one-dimensional viscous heat conductive gas flow equation. Part I: A priori estimates and stability”, Sov. J. Numer. Anal. Math. Modelling, 2:3 (1987), 159–178 | DOI | MR | Zbl

[11] Amosov A.A., Zlotnik A.A., “A study of finite-difference method for the one-dimensional viscous heat conductive gas flow equation. Part II: Error estimates and realization”, Sov. J. Numer. Anal. Math. Modelling, 2:4 (1987), 239–258 | DOI | MR | Zbl

[12] Amosov A.A., Zlotnik A.A., “Two-level finite-difference schemes for one-dimensional equations of magnetic gas dynamics (viscous heat-conducting case)”, Sov. J. Numer. Anal. Math. Modelling, 4:3 (1989), 179–197 | DOI | MR | Zbl

[13] Zlotnik A.A., “Kvazigazodinamicheskaya sistema uravnenii s obschimi uravneniyami sostoyaniya”, Dokl. RAN, 431:5 (2010), 605–609 | Zbl

[14] Zlotnik A.A., “O kvazigazodinamicheskoi sisteme uravnenii s obschimi uravneniyami sostoyaniya i istochnikom tepla”, Matem. modelirovanie, 22:7 (2010), 53–64 | MR | Zbl

[15] Elizarova T.L, Shilnikov E.V., “Vozmozhnosti kvazigazodinamicheskogo algoritma dlya chislennogo modelirovaniya techenii gaza”, Zh. vychisl. matem. i matem. fiz., 49:3 (2009), 549–566 | MR | Zbl

[16] Elizarova T.G., Shilnikov E.V., “Analiz vychislitelnykh svoistv kvazigazodinamicheskogo algoritma na primere resheniya uravnenii Eilera”, Zh. vychisl. matem. i matem. fiz., 49:11 (2009), 1953–1969 | MR | Zbl

[17] Hairer E., Lubich C., Wanner G., Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, 2nd ed., Springer, Berlin, 2006 | MR

[18] Amosov A.A., Zlotnik A.A., “Raznostnaya skhema dlya uravnenii odnomernogo dvizheniya vyazkogo barotropnogo gaza”, Vychisl. protsessy i sistemy, 4, ed. G.I. Marchuk, Nauka, M., 1986, 192–218

[19] 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