A special difference scheme for solving stiff boundary value problems of convective-diffusion transfer

V. G. Zverev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 90 (2024), pp. 5-17

Résumé

The convective-diffusion transfer equation is often found in problems of hydromechanics and heat and mass transfer. The dominance of convection over diffusion and the change in sign of the coefficient at the first derivative lead to the formation of boundary and internal layers with high gradients of the function. This creates serious difficulties in numerical analysis of the problem using traditional difference schemes. The traditional method of approximating the first derivative using central differences at high Peclet numbers can lead to oscillations and violate the monotonicity of the numerical solution. To avoid this problem, it is necessary to significantly reduce the size of grid cells in narrow areas with large gradients of the unknown function. The use of one-sided differences significantly smears the desired solution, due to the viscosity of the scheme, and leads to loss of accuracy. The practical need to solve stiff boundary value problems requires the development and use of computational technologies that guarantee monotonicity, accuracy, and cost-effectiveness in numerical analysis. In this paper, a new special difference scheme is proposed for the numerical solution of a stiff equation of convective-diffusion transfer. The dominant convective term is eliminated from explicit consideration by transforming the equation into self-adjoined form, which permits the use of well-known numerical approximation techniques. The control volume method is used to construct a difference analogue of a differential equation on a three-point template. The resulting scheme is monotonic and conservative. The test examples show great possibilities of the proposed difference scheme for large Peclet numbers on coarse grids in solving stiff boundary value problems of convective diffusion transfer.

Détail

Mots-clés : convective-diffusion transfer
Keywords: difference scheme, control volume method, three-point template, solution's monotonicity.

Bibliographie

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

[2] Patankar S., Chislennye metody resheniya zadach teplomassoobmena i dinamiki zhidkosti, Energoatomizdat, M., 1984

[3] Semenova A.A., Starchenko A.V., “Raznostnaya skhema dlya nestatsionarnogo uravneniya perenosa, postroennaya s ispolzovaniem lokalnykh vesovykh ingerpolyatsionnykh kubicheskikh splainov”, Vestnik Tomskogo gosudarstvennogo universigeta. Matematika i mekhanika, 2017, no. 49, 61–74

[4] Voevodin A. F., “Metod sopryazhennykh operatorov dlya resheniya kraevykh zadach dlya obykno vennykh differentsialnykh uravnenii vtorogo poryadka”, Sibirskii zhurnal vychislitelnoi matematiki, 15:3 (2012), 250–260

[5] Dulan E., Miller Dzh., Shilders U., Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem, Mir, M., 1983

[6] Bagaev B.M., Karepova E.D., Shaidurov V.V., Setochnye metody resheniya zadach s pogranichnym sloem, v. 2, Nauka, Novosibirsk, 2001, 224 pp. | MR

[7] Zadorin A.I., Raznostnye skhemy dlya zadach s pogranichnym sloem, OmGU, Omsk, 2002, 118 pp.

[8] Liseikin V.D., Karasulji'c S., Paasonen V.I., Numerical Grids and High-Order Schemes for Problems with Boundary and Interior Layers, IPC Novosibirsk State University, Novosibirsk, 2021 | MR

[9] Zverev V.G., Chislennye metody resheniya zadach s pogranichnym sloem, Nauka, Novosibirsk, 2017

[10] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1976

[11] Buleev N.I., Prostranstvennaya model turbulentnogo obmena, Nauka, M., 1989 | MR

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

[13] Grishin A.M., Bertsun V.N., Zinchenko V.I., Iteratsionno-interpolyatsionnyi metod i ego prilozheniya, Izd-vo Tom. un-ta, Tomsk, 1981 | MR

[14] Buleev N.I., Timukhin G.I., “O chislennom reshenii uravnenii gidrodinamiki dlya ploskogo potoka vyazkoi neszhimaemoi zhidkosti”, Izvestiya SO AN SSSR. Ser. tekhn. nauk, 1969, no. 1, 14–24 | MR

[15] Ilin A.M., “Raznostnaya skhema dlya differentsialnogo uravneniya s malym parametrom pri starshei proizvodnoi”, Matematichekie zametki, 6:2 (1969), 237–248 | Zbl

[16] Zverev V.G., Goldin V.D., “Raznostnaya skhema dlya resheniya konvektivno-diffuzionnykh zadach teplomassoobmena”, Vychislitelnye tekhnologii, 7:6 (2002), 24–37 | MR | Zbl

[17] Zverev V.G., “Ob odnoi spetsialnoi raznostnoi skheme dlya resheniya kraevykh zadach teplomassoobmena”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 43:2 (2003), 265–278 | MR | Zbl