On stability in $\ell_p$ of some difference schemes for the transport equation

A. Sh. Akysh (Akishev)

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 5 (2002) no. 3, pp. 199-214.

Résumé

In the present work, the stability in the space $\ell_p$, $1$, for a wide class of difference analogs of kinetic transport as well as for the Carleman nonlinear system in the Baltazar equation theory has been proved. The stability in the norm of the space $\ell_p$ results, as a particular case, in the stability in $\ell_2$, which coincides with the stability in the energy space, and at $p=\infty$, with the norm in the space $C$. In this case, the result is gained in the manner similar to the methods of obtaining a priori estimations in the norm of the space $L_p$ for differential problems by themselves.

Bibliographie

[1] Rikhtmaier R., Morton K., Raznostnye metody resheniya kraevykh zadach, Mir, M., 1972

[2] Marchuk G. I., Yanenko N. N., “Reshenie mnogomernogo kineticheskogo uravneniya metodom rasschepleniya”, DAN SSSR, 157:6 (1964), 590–596

[3] Marchuk G. I., Sultangazin U. M., “K obosnovaniyu metoda rasschepleniya dlya uravneniya perenosa izlucheniya”, Zhurn. vychisl. matem. i mat. fiziki, 5:5 (1965), 590–596

[4] Penenko V. V., Sultangazin U. M., Balash B. A., “Reshenie kineticheskogo uravneniya metodom rasschepleniya”, Vychislitelnye metody v teorii perenosa, Atomizdat, M., 1969 | MR

[5] Smelov V. V., Lektsii po teorii perenosa neitronov, Atomizdat, M., 1978 | MR

[6] Godunov S. K., Ryabenkii B. C., Raznostnye skhemy, Nauka, M., 1973 | Zbl

[7] Samarskii A. A., Gulin A. V., Ustoichivost raznostnykh skhem, Nauka, M., 1973 | Zbl

[8] Marchuk G. I., Metody rasschepleniya, Nauka, M., 1988 | MR

[9] Serdyukova S. I., “Issledovanie ustoichivosti v $C$ yavnykh raznostnykh skhem s postoyannymi deistvitelnymi koeffitsientami, ustoichivykh v $l_2$”, Zhurn. vychisl. matem. i mat. fiziki, 3:2 (1963), 365–370 | MR | Zbl

[10] Konovaltsev I. V., “Ustoichivost v $C$ i $L_p$ dvukhsloinykh raznostnykh skhem dlya parabolicheskikh uravnenii”, Zhurn. vychisl. matem. i mat. fiziki, 8:4 (1968), 465–469

[11] Sakhanov N. N., Bulebaev A. E., Smagulov Sh. S., O summirovanii s vesom resheniya dvuchlennogo differentsialnogo uravneniya chetvertogo poryadka na ravnomernoi setke, Dep. v Kaz. NIINKI, Almaty, 23.12.92, No 3965-K92 | Zbl

[12] Mysovskikh I. P., Interpolyatsionnye kubaturnye formuly, Nauka, M., 1981 | MR | Zbl

[13] Akishev A. Sh., “Ob ustoichivosti nekotorykh raznostnykh skhem dlya uravneniya perenosa v prostranstve $l_p(G_h)$, $1\le\infty$”, DAN RK, 3 (1994), 3–16

[14] Akishev A. Sh., “Ob ustoichivosti raznostnykh skhem v prostranstve $l_p$”, Tez. dokl. yubileinoi nauchnoi konferentsii, posvyaschennoi 50-letiyu razvitiya matematiki v AN Kazakhstana (Almaty, sentyabr, 25–29), Almaty, 1995, 22

[15] Marchuk G. I., Metody rascheta yadernykh reaktorov, Gosatomizdat, M., 1961

[16] Karleman T., Matematicheskie zadachi kineticheskoi teorii gazov, IL, M., 1960

[17] Platkowski T., Illner R., “Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory”, SIAM Rev., 30:2 (1988), 213–255 | DOI | MR | Zbl

[18] Akishev A. Sh., Sultangazin A. U., “Novye apriornye otsenki resheniya dlya nelineinykh sistem uravneniya Karlemana”, Vestnik AN KazSSR, 1991, no. 11, 40–47 | MR

[19] Akishev A. Sh., Sultangazin A. U., “Ob ustoichivosti raznostnykh skhem dlya sistem nelineinykh uravnenii Karlemana”, Uslovno-korrektnye zadachi matematicheskoi fiziki i analiza, Nauka, Alma-Ata, 1992, 17–27

[20] Akishev A. Sh., Ustoichivost raznostnykh skhem v prostranstve $l_p$, Problemy vychislitelnoi matematiki i informatsionnykh tekhnologii: Materialy mezhdunarodnoi nauchno-prakticheskoi konferentsii (25–26 marta), Almaty, 1999, 46