Geodesic
Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations
Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 5, pp. 577-586 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon^2$, $\varepsilon \in (0,1]$, multiplying the highest-order derivative in the equation, an initial-boundary value Dirichlet problem is considered. For this problem, a standard difference scheme constructed by using monotone grid approximations of the differential problem on uniform grids, is studied in the presence of computer perturbations. Perturbations of grid solutions are studied, which are generated by computer perturbations, i.e., the computations on a computer. The conditions imposed on admissible computer perturbations are obtained under which the accuracy of the perturbed computer solution is the same by order as the solution of an unperturbed difference scheme, i.e., a standard scheme in the absence of perturbations. The schemes of this type with controlled computer perturbations belong to computer difference schemes, also named reliable difference schemes.
Keywords: initial–boundary value problem, singularly perturbed parabolic equation, standard difference scheme, uniform grid, computer difference scheme.
Mots-clés : reaction-diffusion equation, computer perturbations 
@article{MAIS_2016_23_5_a6,
     author = {G. I. Shishkin},
     title = {Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {577--586},
     year = {2016},
     volume = {23},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2016_23_5_a6/}
}
TY  - JOUR
AU  - G. I. Shishkin
TI  - Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2016
SP  - 577
EP  - 586
VL  - 23
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MAIS_2016_23_5_a6/
LA  - ru
ID  - MAIS_2016_23_5_a6
ER  - 
%0 Journal Article
%A G. I. Shishkin
%T Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations
%J Modelirovanie i analiz informacionnyh sistem
%D 2016
%P 577-586
%V 23
%N 5
%U http://geodesic.mathdoc.fr/item/MAIS_2016_23_5_a6/
%G ru
%F MAIS_2016_23_5_a6
G. I. Shishkin. Computer difference scheme for a singularly perturbed reaction-diffusion equation in the presence of perturbations. Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 5, pp. 577-586. http://geodesic.mathdoc.fr/item/MAIS_2016_23_5_a6/

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

[2] Shishkin G. I., “Computer Difference Scheme for a Singularly Perturbed Convection-Diffusion Equation”, Computational Mathematics and Mathematical Physics, 54:8 (2014), 1221–1233 | DOI | DOI | MR | Zbl

[3] Shishkin G. I., “Standard Scheme for a Singularly Perturbed Parabolic Convection-Diffusion Equation under Computer Perturbation”, Doklady Mathematics, 91:3 (2015), 273–276 | DOI | DOI | MR | Zbl

[4] Shishkin G. I., “Difference Scheme for a Singularly Perturbed Parabolic Convection-Diffusion Equation in the Presence of Perturbations”, Computational Mathematics and Mathematical Physics, 55:11 (2015), 1876–1892 | DOI | MR | Zbl

[5] Shishkin G. I., Shishkina L. P., Difference Methods for Singular Perturbation Problems, CRC Press, Boca Raton, 2009 | MR | Zbl

[6] Shishkin G. I., “Grid approximation of singularly perturbed equations with convective terms under perturbation of data”, Computational Mathematics and Mathematical Physics, 41:5 (2001), 649–664 | MR | Zbl

[7] Shishkin G. I., “Conditioning of finite difference schemes for a singularly perturbed convec-tion-diffusion parabolic equation”, Computational Mathematics and Mathematical Physics, 48:5 (2008), 769–785 | DOI | MR | Zbl