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The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition
G. I. Shishkin Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia
Abstract:
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.
Key words:
singularly perturbed boundary value problem, parabolic reaction-diffusion equation, piecewise continuous initial condition, grid approximation, method of additive splitting of singularities, special grids, $\varepsilon$-uniform convergence, Richardson technique.
UDC:
519.633 Received: 20.10.2008
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