Geodesic
Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 2, pp. 295-306 Cet article a éte moissonné depuis la source Math-Net.Ru

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A stationary convection-diffusion problem with a small parameter multiplying the highest derivative is considered. The problem is discretized on a uniform rectangular grid by the central-difference scheme. A new class of two-step iterative methods for solving this problem is proposed and investigated. The convergence of the methods is proved, optimal iterative methods are chosen, and the rate of convergence is estimated. Numerical results are presented that show the high efficiency of the methods. 
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     title = {Two-step iterative methods for solving the stationary convection-diffusion equation with a~small parameter at the highest derivative on a~uniform grid},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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Zh. Zh. Bai; L. A. Krukier; T. S. Martynova. Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 2, pp. 295-306. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_2_a9/

[1] Samarskii A. A., Vabischevich P. N., Chislennye metody resheniya zadach konvektsii-diffuzii, Editorial URSS, M., 1999

[2] Morton K. W., Numerical solution of convection-diffusion problems, Chapmen-Hall, London, 1996 | MR | Zbl

[3] Zhang J., “Preconditioned iterative methods and finite difference schemes for convection-diffusion”, Appl. Math. and Comput., 109 (2000), 11–30 | DOI | MR | Zbl

[4] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennykh ellipticheskikh uravnenii s konvektivnymi chlenami pri nalichii razlichnykh tipov pogranichnykh sloev”, Zh. vychisl. matem. i matem. fiz., 45:1 (2005), 110–125 | MR | Zbl

[5] Bakhvalov N. S., “K optimizatsii metodov resheniya kraevykh zadach pri nalichii pogranichnogo sloya”, Zh. vychisl. matem. i matem. fiz., 9:4 (1969), 841–859 | Zbl

[6] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984 | MR | Zbl

[7] Krukier L. A., “Neyavnye raznostnye skhemy i iteratsionnyi metod ikh resheniya dlya odnogo klassa sistem kvazilineinykh uravnenii”, Izv. vuzov. Matematika, 1979, no. 7, 41–52 | MR | Zbl

[8] Krukier L. A., Martynova T. S., “O vliyanii formy zapisi uravneniya konvektsii-diffuzii na skhodimost metoda verkhnei relaksatsii”, Zh. vychisl. matem. i matem. fiz., 39:11 (1999), 1821–1827 | MR | Zbl

[9] Taussky O., “Positive-definite matrices and their role in the study of the characteristic roots of general matrices”, Advances Math., 2 (1968), 175–186 | DOI | MR | Zbl

[10] Krukier L. A., Chikina L. G., Belokon T. V., “Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations”, Appl. Numer. Math., 41 (2002), 89–105 | DOI | MR | Zbl

[11] Wang L., Bai Z.-Z., “Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts”, BIT Numer. Math., 44 (2004), 363–386 | DOI | MR | Zbl

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

[13] Bochev M. A., Krukier L. A., “Ob iteratsionnom reshenii silno nesimmetrichnykh sistem lineinykh algebraicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 37:11 (1997), 1283–1293 | MR | Zbl

[14] Bai Z.-Z., Sun J.-C., Wang D.-R., “A unified framework for the construction of various matrix multispliting iterative methods for large sparse system of linear equations”, Comput. Math. and Appl., 32:12 (1996), 51–76 | DOI | MR | Zbl

[15] Belokon T. V., “Ispolzovanie poperemenno-treugolnykh kososimmetrichnykh pereobuslavlivatelei pri reshenii silno nesimmetrichnykh sistem lineinykh algebraicheskikh uravnenii variatsionnym metodom”, X Vseros. shkola-seminar “Sovrem. probl. matem. modelirovaniya”, v. 2, Izd-vo RGU, Rostov-na-Donu, 2004, 60–71

[16] Krukier L. A., “Matematicheskoe modelirovanie protsessov perenosa v neszhimaemykh sredakh s preobladayuschei konvektsiei”, Matem. modelirovanie, 9:2 (1997), 4–12 | MR | Zbl