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 Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 5, Pages 840–856 (Mi zvmmf4689)

Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry

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 boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is specified on the boundary of the domain. The Laplace operator in the differential equation involves a perturbation parameter $\varepsilon^2$, where $\varepsilon$ takes arbitrary values in the half-open interval (0, 1]. When $\varepsilon\to0$, the solution of such a problem has a parabolic boundary layer in a neighborhood of the boundary. Using the integro-interpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge $\varepsilon$-uniformly at a rate of $O(N^{-2}\ln^2N+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of the mesh points in the radial and time variables, respectively.

Key words: boundary value problem, parabolic reaction-diffusion equation, perturbation parameter, parabolic boundary layer, conservative finite difference scheme, piecewise uniform grid, flux grid, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:5, 810–826

Bibliographic databases:

Document Type: Article
UDC: 519.633
Revised: 12.11.2008

Citation: G. I. Shishkin, L. P. Shishkina, “Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry”, Zh. Vychisl. Mat. Mat. Fiz., 49:5 (2009), 840–856; Comput. Math. Math. Phys., 49:5 (2009), 810–826

Citation in format AMSBIB
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This publication is cited in the following articles:
1. G. I. Shishkin, L. P. Shishkina, “A conservative difference scheme for a singularly perturbed elliptic reaction-diffusion equation: approximation of solutions and derivatives”, Comput. Math. Math. Phys., 50:4 (2010), 633–645
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