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

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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 ɛ2, where ɛ takes arbitrary values in the half-open interval (0, 1]. When ɛ → 0, 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 ɛ-uniformly at a rate of O(N −2ln2 N + N −10 ), where N + 1 and N 0 + 1 are the numbers of the mesh points in the radial and time variables, respectively.

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Authors and Affiliations

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

    G. I. Shishkin & L. P. Shishkina

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  1. G. I. Shishkin
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  2. L. P. Shishkina
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Correspondence to G. I. Shishkin.

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Original Russian Text © G.I. Shishkin, L.P. Shishkina, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 5, pp. 840–856.

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Shishkin, G.I., Shishkina, L.P. Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry. Comput. Math. and Math. Phys. 49, 810–826 (2009). https://doi.org/10.1134/S0965542509050078

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  • DOI: https://doi.org/10.1134/S0965542509050078

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