A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

G. I. Shishkin ; L. P. Shishkina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 12, pp. 2113-2133

Résumé

For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon$, where $\varepsilon\in(0,1]$, the grid approximation of the Dirichlet problem on a rectangular domain in the $(x,t)$-plane is examined. For small $\varepsilon$, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of $\varepsilon$-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges $\varepsilon$-uniformly in the maximum norm at the rate of $O(N^{-2}\ln^2N+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme converges $\varepsilon$-uniformly at the rate of $O(N^{-4}\ln^4N+N_0^{-2})$. For fixed values of the parameter, the convergence rate is $O(N^{-4}+N_0^{-2})$.