Geodesic
Adaptive grids and high-order schemes for solving singularly-perturbed problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 77-92 
V. D. Liseikin; V. I. Paasonen. Adaptive grids and high-order schemes for solving singularly-perturbed problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 77-92. http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a6/
@article{SJVM_2021_24_1_a6,
     author = {V. D. Liseikin and V. I. Paasonen},
     title = {Adaptive grids and high-order schemes for solving singularly-perturbed problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {77--92},
     year = {2021},
     volume = {24},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a6/}
}
TY  - JOUR
AU  - V. D. Liseikin
AU  - V. I. Paasonen
TI  - Adaptive grids and high-order schemes for solving singularly-perturbed problems
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2021
SP  - 77
EP  - 92
VL  - 24
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a6/
LA  - ru
ID  - SJVM_2021_24_1_a6
ER  - 
%0 Journal Article
%A V. D. Liseikin
%A V. I. Paasonen
%T Adaptive grids and high-order schemes for solving singularly-perturbed problems
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2021
%P 77-92
%V 24
%N 1
%U http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a6/
%G ru
%F SJVM_2021_24_1_a6

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

Layer-resolving grids remain an important element of comprehensive software codes when solving real-life problems with layers of singularities as they can substantially enhance the efficiency of computer-resource utilization. This paper describes an explicit approach to generating layer-resolving grids which is aimed at application of difference schemes of an arbitrary order. The approach proposed is based on estimates of derivatives of solutions to singularly-perturbed problems and is a generalization of the approach developed for the first order schemes. The layer-resolving grids proposed are suitable to tackle problems with exponential-, power-, logarithmic-, and mixed-type boundary and interior layers. Theoretical results have been confirmed by the numerical experiments on a number of test problems with such layers; the results were compared to those obtained with difference schemes of different orders of accuracy.

[1] “On the Optimization of the Methods for Solving Boundary Value Problems in the Presence of a Boundary Layer”, J. Comput. Maths. Math. Phys., 9:4 (Bakhvalov N.S.), 139–166 | MR | Zbl

[2] Shishkin G.I., “A Difference scheme for a singularly perturbed equation of parabolic type with a discontinuous initial condition”, Dokl. Math., 37:3 (1988), 792–796 | MR | Zbl

[3] Liseikin V.D., Paasonen V.I., “Compact difference schemes and layer-resolving grids for numerical modeling of problems with boundary and interior layers”, Numerical Analysis and Applications, 12:1 (2019), 37–50 | DOI | MR | MR

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

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

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

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

[8] Liseikin V.D., “On the numerical solution of singularly perturbed equations with a turning point”, J. Comput. Maths. Math. Phys., 26:6 (1986), 133–139 | DOI | MR | MR

[9] P. Vulanovic̀, “Mesh construction for numerical solution of a type of singular perturbation problems”, Numer. Meth. Approx. Theory, University of Nis̆, 1984, 137–142 | MR

[10] P.Ya. Polubarinova-Kochina, Teoriya dvizheniya gruntovykh vod, Nauka, M., 1977 | MR

[11] K. I. Zamaraev, R. F. Khairutdinov, V. P. Zhdanov, Tunnelirovanie elektronov v khimii: Khimicheskie reaktsii na bol'shikh rasstoyaniyakh, Nauka, Novosibirsk, 1985