Geodesic
On the solution of evolution equations based on multigrid and explicit iterative methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 8, pp. 1305-1319 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two schemes for solving initial–boundary value problems for three-dimensional parabolic equations are studied. One is implicit and is solved using the multigrid method, while the other is explicit iterative and is based on optimal properties of the Chebyshev polynomials. In the explicit iterative scheme, the number of iteration steps and the iteration parameters are chosen as based on the approximation and stability conditions, rather than on the optimization of iteration convergence to the solution of the implicit scheme. The features of the multigrid scheme include the implementation of the intergrid transfer operators for the case of discontinuous coefficients in the equation and the adaptation of the smoothing procedure to the spectrum of the difference operators. The results produced by these schemes as applied to model problems with anisotropic discontinuous coefficients are compared. 
@article{ZVMMF_2015_55_8_a3,
     author = {V. T. Zhukov and N. D. Novikova and O. B. Feodoritova},
     title = {On the solution of evolution equations based on multigrid and explicit iterative methods},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1305--1319},
     year = {2015},
     volume = {55},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_8_a3/}
}
TY  - JOUR
AU  - V. T. Zhukov
AU  - N. D. Novikova
AU  - O. B. Feodoritova
TI  - On the solution of evolution equations based on multigrid and explicit iterative methods
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2015
SP  - 1305
EP  - 1319
VL  - 55
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_8_a3/
LA  - ru
ID  - ZVMMF_2015_55_8_a3
ER  - 
%0 Journal Article
%A V. T. Zhukov
%A N. D. Novikova
%A O. B. Feodoritova
%T On the solution of evolution equations based on multigrid and explicit iterative methods
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2015
%P 1305-1319
%V 55
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_8_a3/
%G ru
%F ZVMMF_2015_55_8_a3
V. T. Zhukov; N. D. Novikova; O. B. Feodoritova. On the solution of evolution equations based on multigrid and explicit iterative methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 8, pp. 1305-1319. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_8_a3/

[1] Fedorenko R. P., “Relaksatsionnyi metod resheniya raznostnykh ellipticheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 1:5 (1961), 922–927 | MR | Zbl

[2] Fedorenko R. P., “Iteratsionnye metody resheniya raznostnykh ellipticheskikh uravnenii”, Uspekhi matem. nauk, XXVIII:2(170) (1973), 129–195 | Zbl

[3] Fedorenko R. P., Vvedenie v vychislitelnuyu fiziku, MFTI, M., 1994

[4] Zhukov V. T., Novikova N. D., Feodoritova O. B., “O mnogosetochnom i yavno-iteratsionnom metodakh dlya parabolicheskikh uravnenii”, Preprinty IPM im. M. V. Keldysha, 2014, 028, 36 pp. http://library.keldysh.ru/preprint.asp?id=2014-28 | Zbl

[5] Zhukov V. T., Novikova N. D., Feodoritova O. B., “O primenenii mnogosetochnogo i yavno-iteratsionnogo metodov k resheniyu parabolicheskikh uravnenii s anizotropnymi razryvnymi koeffitsientami”, Preprinty IPM im. M. V. Keldysha, 2014, 085, 24 pp. http://library.keldysh.ru/preprint.asp?id=2014-85

[6] Zhukov V. T., Novikova N. D., Feodoritova O. B., “Parallelnyi mnogosetochnyi metod dlya raznostnykh ellipticheskikh uravnenii”, Matem. modelirovanie, 26:1 (2014), 55–68 | Zbl

[7] Zhukov V. T., Novikova N. D., Feodoritova O. B., “Mnogosetochnyi metod dlya anizotropnykh uravnenii diffuzii na osnove adaptatsii chebyshevskikh sglazhivatelei”, Matem. modelirovanie, 26:9 (2014), 126–140 | Zbl

[8] Zhukov V. T., Novikova N. D., Feodoritova O. B., “Mnogosetochnyi metod dlya ellipticheskikh uravnenii s anizotropnymi razryvnymi koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 55:7 (2015), 1168–1182 | DOI

[9] Zhukov V. T., “O yavnykh metodakh chislennogo integrirovaniya dlya parabolicheskikh uravnenii”, Matem. modelirovanie, 22:10 (2010), 127–158 | MR | Zbl

[10] Lokutsievskii V. O., Lokutsievskii O. V., “O chislennom reshenii kraevykh zadach dlya uravnenii parabolicheskogo tipa”, Dokl. AN SSSR, 291:3 (1986), 540–544 | MR

[11] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978 | MR

[12] Gelfand I. M., Lokutsievskii O. V., “O raznostnykh skhemakh dlya resheniya uravneniya teploprovodnosti”: Godunov S. K., Ryabenkii B. C., Vvedenie v teoriyu raznostnykh skhem, Fizmatgiz, M., 1962

[13] Babenko K. I., Osnovy chislennogo analiza, NITs RKhD, M.–Izhevsk, 2002

[14] Lyusternik L. A., “O raznostnykh approksimatsiyakh operatora Laplasa”, Uspekhi matem. nauk, IX:2(60) (1954), 3–66 | Zbl

[15] Shvedov A. S., Zhukov V. T., “Explicit iterative difference schemes for parabolic equations”, Russian J. Numer. Anal. Math. Modelling, 13:2 (1998), 133–148 | DOI | MR | Zbl

[16] Chebyshev P. L., Sochineniya, v. 1, Voprosy o naimenshikh velichinakh, svyazannye s priblizhennym predstavleniem funktsii, SPb., 1899

[17] Lebedev V. I., Finogenov S. A., “O poryadke vybora iteratsionnykh parametrov v chebyshevskom tsiklicheskom iteratsionnom metode”, Zh. vychisl. matem. i matem. fiz., 11:2 (1971), 425–438 | MR | Zbl

[18] Gantmakher F. R., Teoriya matrits, Nauka, M., 1966 | MR

[19] Trottenberg U., Oosterlee C. W., Schuller A., Multigrid, Academic Press, 2001 | MR | Zbl

[20] http://www.kiam.ru/MVS/resourses/k100.html