Geodesic
Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 1, pp. 41-56.

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

This paper realizes a symbiosis of two approaches to the numerical solution of second order ODEs with a small parameter having singularities such as interior and boundary layers, namely, the application of both compact schemes of high orders and layer-resolving grids. The generation of layer-resolving grids, based on estimates of solution derivatives and formulations of coordinate transformations eliminating solution singularities, is a generalization of the methodology early developed for the first order scheme. This paper presents the formulas of the coordinate transformations and numerical experiments for the schemes of the first, second, and third orders on uniform and layer-resolving grids for the equations with boundary, interior, exponential and power layers of the first and second scales. The experiments conducted confirm the uniform convergence of the numerical solutions of equations with the help of compact schemes of high orders on the layer-resolving grids. By using the transfinite interpolation methodology or numerical solutions to the Beltrami and diffusion equations in a control metric, built by the coordinate transformations eliminating the solution singularities, the developed technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers. 
@article{SJVM_2019_22_1_a3,
     author = {V. D. Liseikin and V. I. Paasonen},
     title = {Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {41--56},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2019_22_1_a3/}
}
TY  - JOUR
AU  - V. D. Liseikin
AU  - V. I. Paasonen
TI  - Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2019
SP  - 41
EP  - 56
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2019_22_1_a3/
LA  - ru
ID  - SJVM_2019_22_1_a3
ER  - 
%0 Journal Article
%A V. D. Liseikin
%A V. I. Paasonen
%T Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2019
%P 41-56
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2019_22_1_a3/
%G ru
%F SJVM_2019_22_1_a3
V. D. Liseikin; V. I. Paasonen. Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 1, pp. 41-56. http://geodesic.mathdoc.fr/item/SJVM_2019_22_1_a3/

[1] Liseikin V. D., “Ocenki proizvodnykh resheniya differencial'nykh uravneniy s pogranichnymi i vnutrennimi sloyami”, Sibirskiy matematicheskiy zhurnal, 33:6 (1992), 106–117 | MR

[2] Liseikin V. D., Layer Resolving Grids and Transformations for Singular Perturbation Problems, VSP, Utrecht, 2001

[3] Liseikin V. D., Grid Generation Methods, Third ed., Springer, Berlin, 2017 | MR

[4] Paasonen V. I., “Kompaktnye skhemy tret'ego poryadka tochnosti na neravnomernykh adaptivnykh setkakh”, Vychislitel'nye tekhnologii, 20:2 (2015), 56–64 | Zbl

[5] Paasonen V. I., “Skhema tret'ego poryadka approksimacii na neravnomernoy setke dlya uravneniy Nav'e-Stoksa”, Vychislitel'nye tekhnologii, 5:5 (2000), 78–85 | MR | Zbl

[6] Glukhovskiy A. S., Paasonen V. I., “Kompaktnye raznostnye skhemy dlya uravneniy Nav'e-Stoksa na neravnomernykh setkakh”, Marchukovskie nauchnye chteniya 2017. Tr. Mezhdunar. konf. “Vychislitel'naya i prikladnaya matematika 2017” (25–30 iyunya 2017 g.), Izd-vo IVMiMG SO RAN, Novosibirsk, 2017, 211–217

[7] Liseikin V. D., “O chislennom reshenii uravneniy so stepennym pogranichnym sloem”, Zhurn. vychisl. matem. i mat. fiziki, 26:12 (1986), 1813–1820 | MR

[8] Bakhvalov N. S., “Ob optimizacii metodov chislennogo resheniya kraevykh zadach s pogranichnymi sloyami”, Zhurn. vychisl. matem. i mat. fiziki, 9:4 (1969), 842–859

[9] Shishkin G. I., “Raznostnaya skhema dlya singulyarno vozmushchennogo uravneniya parabolicheskogo tipa s razryvnym nachal'nym usloviem”, DAN SSSR, 37 (1988), 792–796 | Zbl

[10] Liseikin V. D., “O chislennom reshenii singulyarno vozmushchennykh uravneniy s tochkami povorota”, Zhurn. vychisl. matem. i mat. fiziki, 24:12 (1984), 1812–1818 | MR