Geodesic
A numerical method for solving diffusion-type equations based on a multigrid method
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 8, pp. 1438-1461
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A new numerical algorithm based on multigrid methods is proposed for solving equations of the parabolic type. Theoretical error estimates are obtained for the algorithm as applied to a two-dimensional initial-boundary value model problem for the heat equation. The good accuracy of the algorithm is demonstrated using model problems including ones with discontinuous coefficients. As applied to initial-boundary value problems for diffusion equations, the algorithm yields considerable savings in computational work compared to implicit schemes on fine grids or explicit schemes with a small time step on fine grids. A parallelization scheme is given for the algorithm. 
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M. E. Ladonkina; O. Yu. Milyukova; V. F. Tishkin. A numerical method for solving diffusion-type equations based on a multigrid method. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 8, pp. 1438-1461. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_8_a7/

[1] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[2] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody http://phys.spb.ru/Stud/Books/index.php

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

[4] Bakhvalov N. S., “O skhodimosti odnogo relaksatsionnogo metoda pri estestvennykh ogranicheniyakh na ellipticheskii operator”, Zh. vychisl. matem. i matem. fiz., 6:5 (1966), 861–883 | Zbl

[5] Ladonkina M. E., Milyukova O. Yu., Tishkin V. F., “Odin chislennyi algoritm dlya uravnenii diffuzionnogo tipa na osnove mnogosetochnykh metodov”, Matem. modelirovanie, 19:4 (2007), 71–89 | MR

[6] Ladonkina M. E., Milyukova O. Yu., Tishkin V. F., “Ispolzovanie mnogosetochnykh metodov dlya resheniya uravnenii diffuzionnogo tipa”, VANT. Ser. Matem. modelirovanie fiz. protsessov, 2008, no. 1, 4–19 | MR

[7] Ladonkina M. E., Milyukova O. Yu., Tishkin V. F., “Chislennyi algoritm resheniya diffuzionnykh uravnenii na osnove mnogosetochnykh metodov”, Zh. vychisl. matem. i matem. fiz., 49:3 (2009), 518–541 | MR

[8] Ladonkina M. E., Milyukova O. Yu., Tishkin V. F., “Konservativnye skhemy dlya resheniya uravnenii diffuzionnogo tipa na osnove ispolzovaniya mnogosetochnykh metodov”, Tr. Srednevolzhskogo matem. ob-va (Saransk), 10:2 (2008), 21–44

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

[10] Saad Y., Iterative methods for sparse linear systems, PWS Publishing Co.; Int. Tompson Publ. Co., 1995

[11] Godunov S. K., Ryabenkii V. S., Raznostnye skhemy, Nauka, M., 1977 | MR

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

[13] McBryan O. A., Frederikson P. O., Linden J. et al., “Multigrid methods on parallel computers — a survey of recent developments”, Impact Comput. Sci. Engng., 3 (1991), 1–75 | DOI | MR | Zbl

[14] Ritzdorf H., Schüller A., Steckler B., Stüben K., “LiSS — An environment for the parallel multigrid solution of partial differential equation on general domains”, Parallel Comput., 20 (1994), 1559–1570 | DOI

[15] Milyukova O. Yu., “Parallel approximate factorization method for solving discreate elliptic equations”, Parallel Comput., 27 (2001), 1365–1379 | DOI | MR | Zbl

[16] Gustafsson I., “A class of first order factorization methods”, BIT, 18 (1978), 142–156 | DOI | MR | Zbl