Improved difference scheme of the solution decomposition method for a~singularly perturbed reaction-diffusion equation

G. I. Shishkin ; L. P. Shishkina

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 1, pp. 255-271

Résumé

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes convergent uniformly with respect to the perturbation parameter $\varepsilon$, $\varepsilon\in(0,1]$. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-2}\ln^{-2}N)$, where $N+1$ is the number of nodes in the grids used; for fixed values of the parameter $\varepsilon$, the scheme converges at the rate $\mathcal O(N^{-2})$. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-4 }\ln^{-4}N)$.

Détail

Keywords: singularly perturbed boundary value problem, ordinary differential reaction-diffusion equation, asymptotic construction technique, difference scheme of the solution decomposition method, uniform grids, $\varepsilon$-uniform convergence, improved scheme of the solution decomposition method.
Mots-clés : decomposition of a discrete solution, Richardson technique