The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 8, pp. 1416-1436

Résumé

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter $\varepsilon^2$, where $\varepsilon\in(0,1]$. When $\varepsilon$ is small, a boundary and an interior layer (with the characteristic width $\varepsilon$) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed $\varepsilon$, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges $\varepsilon$-uniformly at a rate of $O(N^{-2}\ln^2+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of the mesh points in $x$ and $t$, respectively. Based on the Richardson technique, a scheme that converges $\varepsilon$-uniformly at a rate of $ON^{-3}+N_0^{-2})$ is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in $\varepsilon$-uniformly in $x$ with an order greater than three.