Volume 3, Issue 2
Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation

Int. J. Numer. Anal. Mod. B, 3 (2012), pp. 168-184

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• Abstract

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed to construct difference schemes convergent uniformly with respect to a perturbation parameter ε for ε ∈ (0, 1], i.e., ε-uniformly. This approach is based on a decomposition of the discrete solution into the regular and singular components which are solutions of discrete subproblems considered on uniform grids. Using an asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ε-uniformly in the maximum norm at the rate O(N^{-2}ln^2N), where N + 1 is the number of nodes in the grids used; for fixed values of the parameter ε, the scheme converges at the rate O(N^{-2}). For the constructed scheme, approximations of the regular and singular components to the solution and their derivatives up to the second order are studied. A modified scheme of the solution decomposition method is constructed for which the regular component of the solution and its discrete derivatives converge $\varepsilon$-uniformly in the maximum norm at the rate O(N^{-2}) for ε = o(ln^{-1} N).

• History

Published online: 2012-03

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