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 Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 4, Pages 660–673 (Mi zvmmf156)

Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

G. I. Shishkin, L. P. Shishkina

Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia

Abstract: The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter $\varepsilon^2$, where $\varepsilon$ takes arbitrary values in the interval (0, 1]. When $\varepsilon$ vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to $t$. When $\varepsilon$ tends to zero, a parabolic boundary layer with a characteristic width $\varepsilon$ appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges $\varepsilon$-uniformly at a rate of $O(N^{-2}\ln^2N+N_0^{-1})$, where $N=\min_s N_s$, $N_s+1$ and $N_s+1$ are the numbers of mesh points on the axes $x_s$ and $t$, respectively.

Key words: initial boundary value problem in a rectangle, perturbation parameter $\varepsilon$, system of parabolic reaction-diffusion equations, finite difference approximation, parabolic boundary layer, a priori bounds on the solution and its derivatives, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:4, 627–640

Bibliographic databases:

UDC: 519.633

Citation: G. I. Shishkin, L. P. Shishkina, “Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle”, Zh. Vychisl. Mat. Mat. Fiz., 48:4 (2008), 660–673; Comput. Math. Math. Phys., 48:4 (2008), 627–640

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Shishkin G.I., Shishkina L.P., “Approximation of a system of semilinear singularly perturbed parabolic reaction-diffusion equations on a vertical strip”, International Workshop on Multi-Rate Processes and Hysteresis, Journal of Physics Conference Series, 138, 2008
2. Shishkina L., Shishkin G., “Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction-diffusion equations”, Math. Model. Anal., 14:2 (2009), 211–228
3. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Appl. Math. Comput., 217:8 (2010), 3641–3716
4. Clavero C., Gracia J.L., “Uniformly Convergent Additive Schemes For 2D Singularly Perturbed Parabolic Systems of Reaction-Diffusion Type”, Numer. Algorithms, 80:4 (2019), 1097–1120
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