Geodesic
Monotonicity criteria for difference schemes designed for hyperbolic equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 9, pp. 1638-1667 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Previously formulated monotonicity criteria for explicit two-level difference schemes designed for hyperbolic equations (S. K. Godunov's, A. Harten's (TVD schemes), characteristic criteria) are extended to multileveled, including implicit, stencils. The characteristic monotonicity criterion is used to develop a universal algorithm for constructing high-order accurate nonlinear monotone schemes (for an arbitrary form of the desired solution) based on their analysis in the space of grid functions. Several new fourth-to-third-order accurate monotone difference schemes on a compact three-level stencil and nonexpanding (three-point) stencils are proposed for an extended system, which ensures their monotonicity for both the desired function and its derivatives. The difference schemes are tested using the characteristic monotonicity criterion and are extended to systems of hyperbolic equations. 
@article{ZVMMF_2006_46_9_a7,
     author = {A. S. Kholodov and Ya. A. Kholodov},
     title = {Monotonicity criteria for difference schemes designed for hyperbolic equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1638--1667},
     year = {2006},
     volume = {46},
     number = {9},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_9_a7/}
}
TY  - JOUR
AU  - A. S. Kholodov
AU  - Ya. A. Kholodov
TI  - Monotonicity criteria for difference schemes designed for hyperbolic equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 1638
EP  - 1667
VL  - 46
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_9_a7/
LA  - ru
ID  - ZVMMF_2006_46_9_a7
ER  - 
%0 Journal Article
%A A. S. Kholodov
%A Ya. A. Kholodov
%T Monotonicity criteria for difference schemes designed for hyperbolic equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 1638-1667
%V 46
%N 9
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_9_a7/
%G ru
%F ZVMMF_2006_46_9_a7
A. S. Kholodov; Ya. A. Kholodov. Monotonicity criteria for difference schemes designed for hyperbolic equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 9, pp. 1638-1667. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_9_a7/

[1] Neuman J. von, Richtmyer R. D., “A method for numerical calculation of hydrodynamic shocks”, J. Appl. Phys., 21:1 (1950), 232–237 | DOI | MR | Zbl

[2] Samarskii A. A., “O regulyarizatsii raznostnykh skhem”, Zh. vychisl. matem. i matem. fiz., 7:1 (1967), 62–93 | MR

[3] Chudov L. A., Nekotorye primeneniya raznostnykh metodov v mekhanike zhidkostei i gaza, Dis. $\dots$ dokt. fiz.-matem. nauk, In-t problem mekhan. AN SSSR, M., 1967

[4] Kolgan V. P., “Primenenie operatorov sglazhivaniya v raznostnykh skhemakh vysokogo poryadka tochnosti”, Zh. vychisl. matem. i matem. fiz., 18:5 (1978), 1340–1345 | MR | Zbl

[5] Friedrichs K. O., Hyers D. H., “Symmetrie hyperbolic linear differential equations”, Communs Pure and Appl. Math., 7:2 (1954), 345–392 | DOI | MR | Zbl

[6] Courant R., Isacson E., Rees M., “On the solutions of nonlinear hyperbolic differential equations by finite differences”, Communs Pure and Appl. Math., 5:5 (1952), 243–254 | DOI | MR

[7] Godunov C. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matem. sb., 47(89):3 (1959), 271–306 | MR | Zbl

[8] Lax P. D., “Week solutions nonlinear hyperbolic equations and their numerical computations”, Communs Pure and Appl. Math., 7:1 (1954), 159–193 | DOI | MR | Zbl

[9] Fedorenko P. P., “Primenenie raznostnykh skhem vysokoi tochnosti dlya chislennogo resheniya giperbolicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 2:6 (1962), 1122–1128 | MR | Zbl

[10] Petrov I. B., Kholodov A. C., “O regulyarizatsii razryvnykh chislennykh reshenii uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 24:8 (1984), 1172–1188 | MR | Zbl

[11] Boris I. P., Book D. L., “Flux-corrected transport. I. Shasta a fluid transport algorithm that works”, J. Comput. Phys., 11:1 (1973), 38–69 | DOI | Zbl

[12] Zhmakin A. I., Fursenko A. A., “Ob odnoi monotonnoi raznostnoi skheme skvoznogo scheta”, Zh. vychisl. matem. i matem. fiz., 20:4 (1980), 1021–1031 | MR

[13] Harten A., “High resolution schemes for hyperbolic conservation laws”, J. Comput. Phys., 49:3 (1987), 357–393 | DOI | MR

[14] Kholodov A. C., “Setochno-kharakteristicheskie chislennye metody dlya mnogomernykh zadach mekhaniki sploshnykh sred”, Vopr. kibernetiki, 15, Nauchn. sovet po kompleksnoi probleme “Kibernetika” AN SSSR, M., 1987, 140–163

[15] Kholodov A. C., “Raznostnye skhemy s polozhitelnoi approksimatsiei dlya mnogomernykh sistem uravnenii giperbolicheskogo tipa na neregulyarnykh setkakh”, Ratsionalnoe chisl. modelirovanie v nelineinoi mekhan., Nauka, M., 1990, 49–62 | Zbl

[16] Van Leer B., “Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme”, J. Comput. Phys., 14 (1974), 361–370 | DOI | Zbl

[17] Kholodov Ya. A., “A monotone high-order accuracy schemes for hyperbolic CFD problems”, APS 53rd Meeting Div. Fluid Dynamics, Washington, 2000

[18] Kholodov A. C., “O postroenii raznostnykh skhem s polozhitelnoi approksimatsiei dlya uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 18:6 (1978), 1476–1492 | MR | Zbl

[19] Kholodov A. C., “O postroenii raznostnykh skhem povyshennogo poryadka tochnosti dlya uravnenii giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 20:6 (1980), 1601–1620 | MR | Zbl

[20] Magomedov K. M., Kholodov A. C., Setochno-kharakteristicheskie chislennye metody, Nauka, M., 1988 | MR

[21] Magomedov K. M., Kholodov A. C., “O postroenii raznostnykh skhem dlya uravnenii giperbolicheskogo tipa na osnove kharakteristicheskikh sootnoshenii”, Zh. vychisl. matem. i matem. fiz., 9:2 (1969), 373–386 | MR | Zbl

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

[23] Shokin Yu. I., Yanenko H. H., Metod differentsialnogo priblizheniya. Primenenie k gazovoi dinamike, Nauka, Novosibirsk, 1985 | MR | Zbl

[24] Lax P. D., Wendroff B., “System of conservation laws”, Communs Pure and Appl. Math., 13:2 (1960), 217–237 | DOI | MR | Zbl

[25] MacCormak R. W., The effect of viscosity in hypervelocity impact cratering, AIAA Paper, No 69-354, 1969

[26] Warming R. F., Beam R. M., “Upwind second-order difference schemes and applications to unsteady aerodynamic flow”, Proc. AIAA 2nd Comput. Fluid Dynamics. Conf., Hartford, Connecticut, 1975, 17 | Zbl

[27] Goloviznin V. M., Karabasov S. A., “Nelineinaya korrektsiya skhemy Kabare”, Matem. modelirovanie, 10:12 (1998), 107–123

[28] Tolstykh A. I., Kompaktnye raznostnye skhemy i ikh primenenie v zadachakh aerogidrodinamiki, Nauka, M., 1990 | MR

[29] Courant R., Lax P., “On nonlinear partial differential equations with two independent variables”, Communs Pure and Appl. Math., 2:2–3 (1949), 255–273 | DOI | MR | Zbl

[30] Grudnitskii V. G., Prokhorchuk Yu. A., “Odin priem postroeniya raznostnykh skhem s proizvolnym poryadkom approksimatsii differentsialnykh uravnenii v chastnykh proizvodnykh”, Dokl. AN SSSR, 234:6 (1977), 1249–1252 | MR

[31] Belotserkovskii O. M., Guschin V. A., Konshin V. N., “Metod rasschepleniya dlya issledovaniya techenii stratifitsirovannoi zhidkosti so svobodnoi poverkhnostyu”, Zh. vychisl. matem. i matem. fiz., 27:4 (1987), 594–609 | MR

[32] Vorobev O. V., Kholodov Ya. A., “Ob odnom metode chislennogo integrirovaniya odnomernykh zadach gazovoi dinamiki”, Matem. modelirovanie, 8:1 (1996), 77–92 | MR

[33] Anuchina H. H., “Nekotorye raznostnye skhemy dlya giperbolicheskikh sistem”, Tr. MIAN SSSR, 74, 1966, 5–15 | Zbl

[34] Warming R. F., Kutler P., Lomax H., “Second and third-order noncentered difference schemes for nonlinear hyperbolic equations”, AIAA Journal, 11:2 (1973), 189–196 | DOI | MR | Zbl

[35] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1977 | MR | Zbl

[36] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1977 | MR | Zbl

[37] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Nauka, M., 2001 | MR

[38] Kholodov A. C., Kholodov Ya. A., Stupitskii E. L., Repin A. Yu., “Chislennoe issledovanie povedeniya vysokoenergetichnogo plazmennogo sgustka v verkhnei ionosfere”, Matem. modelirovanie, 16:8 (2004), 3–23 | Zbl