Geodesic
Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 3, pp. 281-297.

Voir la notice de l'article provenant de la source Math-Net.Ru

The first boundary value problem is considered on a strip for a system of two singularly perturbed parabolic equations. The perturbation parameters multiplying the highest derivatives of each of the equations are mutually independent and can take arbitrary values from the interval $(0,1]$. When these parameters equal zero, the system of parabolic equations degenerates into a system of hyperbolic first order equations coupled by the reaction terms. The convective terms (i.e., their components orthogonal to the boundaries of the strip) that are involved in the different equations have the opposite directions (convection with counterflow). This case brings us to the appearance of boundary layers in the neighbourhood of both boundaries of the strip. For this boundary value problem, the difference schemes that converge uniformly with respect to the parameters are constructed here using the condensing mesh method. We also consider the construction of parameter uniform convergent difference schemes for a system of singularly perturbed elliptic equations that degenerate into first order equations if the parameter equals zero. 
@article{SJVM_1998_1_3_a6,
     author = {G. I. Shishkin},
     title = {Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {281--297},
     publisher = {mathdoc},
     volume = {1},
     number = {3},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SJVM_1998_1_3_a6/}
}
TY  - JOUR
AU  - G. I. Shishkin
TI  - Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 1998
SP  - 281
EP  - 297
VL  - 1
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_1998_1_3_a6/
LA  - en
ID  - SJVM_1998_1_3_a6
ER  - 
%0 Journal Article
%A G. I. Shishkin
%T Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 1998
%P 281-297
%V 1
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_1998_1_3_a6/
%G en
%F SJVM_1998_1_3_a6
G. I. Shishkin. Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 3, pp. 281-297. http://geodesic.mathdoc.fr/item/SJVM_1998_1_3_a6/

[1] Samarskii A. A., Theory of Difference Schemes, Nauka, M., 1989 (in Russian) | MR

[2] Marchuk G. I., Methods of Numerical Mathematics, Springer, New York, 1982 | MR

[3] Bakhvalov N. S., “On the optimization of methods for solving boundary value problems in the presence of a boundary layer”, Zh. Vychisl. Mat. Mat. Fiz., 9 (1969), 841–859 (in Russian) | MR | Zbl

[4] Shishkin G. I., Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations, Ural Branch of Russian Academy of Sciences, Ekaterinburg, 1992 (in Russian) | Zbl

[5] Shishkin G. I., “Grid approximation of a singularly perturbed boundary value problem for a quasi-linear elliptic equation degenerating into the first-order equation”, Soviet J. Numer. Anal. Math. Modelling, 6:1 (1991), 61–81 | DOI | MR | Zbl

[6] Shishkin G. I., “Methods of constructing grid approximations for singularly perturbed boundaryvalue problems”, Russian J. Numer. Anal. Math. Modelling, 7:6 (1992), 537–562 | DOI | MR | Zbl

[7] USSR Comp. Maths. Math. Phys., 32 (1992), 467–480 | MR | Zbl

[8] Il'in A. M., “A difference scheme for a differential equation with a small parameter affecting the highest derivative”, Math. Notes, 6:2 (1969), 596–602 | MR

[9] Doolan E. P., Miller J. J. H., and Schilders W. H. A., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Dublin, 1980 | MR

[10] Shishkin G. I., “Grid approximation of singularly perturbed systems of elliptic and parabolic equations with convective terms”, Differents. Uravnenia (to appear) (in Russian)

[11] Brainina Kh. Z. and Neuman Ye. Ya., Solid-Phase Reactions in Electroanalytic Chemistry, Khimiya, M., 1982 (in Russian)

[12] Ladyzhenskaya O. A., Solonnikov V. A., and Ural'tseva N. N., Linear and Quasi-linear Equations of Parabolic Type, Transl. of Math. Monographs, 23, AMS, Providence, RI, 1968 | Zbl

[13] Friedman A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964 | MR | Zbl

[14] Il'in A. M., Kalashnikov A. S., and Oleinik O. A., “Linear second-order equations of parabolic type”, Uspekhi Mat. Nauk, 17:3 (1962), 3–146 (in Russian) | MR

[15] Ladyzhenskaya O. A. and Ural'tseva N. N., Linear and Quasi-linear Equations of Elliptic Type, Nauka, M., 1973 (in Russian) | MR