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A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation
G. I. Shishkin,
L. P. Shishkina Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia
Abstract:
An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge
$\varepsilon$-uniformly in the maximum norm (where
$\varepsilon$ is the perturbation parameter multiplying the highest order derivative,
$\varepsilon\in(0, 1]$). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges
$\varepsilon$-uniformly in the maximum norm at a rate of
$\mathcal{O}(N^{-4}\ln^4N+N_0^{-2})$, where
$N+1$ and
$N_0+1$ are the numbers of nodes in uniform meshes in
$a$ and
$t$, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge
$\varepsilon$-uniformly with an order close to the sixth in
$x$ and equal to the third in
$t$.
Key words:
singularly perturbed initial-boundary value problem, parabolic reaction-diffusion equation, perturbation parameter $\varepsilon$, solution decomposition method, numerical-analytical scheme, improved Richardson difference scheme, $\varepsilon$-uniform convergence, maximum norm.
UDC:
519.633
MSC: Primary
65M06; Secondary
35B25,
35K20,
35K57,
65M12 Received: 31.07.2014
DOI:
10.7868/S0044466915030175
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