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Grid approximation for a singularly perturbed parabolic reaction-diffusion equation with a moving concentrated source
G. I. Shishkin Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
On an axis
$\mathbb R$, we consider an initial value problem for a singularly perturbed parabolic reactiondiffusion equation in the presence of a moving concentrated source. Classical finite difference schemes for such problem converge only under the condition
$\varepsilon\gg N^{-1}+N_0^{-1}$, where
$\varepsilon$ is the singular perturbation parameter, the values
$N$ and
$N_0$ define the number of nodes in the grids with respect to
$x$ (on a segment of unit length) and
$t$. We study schemes on meshes which are locally refined in a neighbourhood of the set
$\gamma^*$, that is, the trajectory of the moving source. It is shown that there are no schemes convergent
$\varepsilon$-uniformly, in particular, for
$\varepsilon=\mathscr O(N^{-2}+N_0^{-2})$, in the class of schemes based on classical approximations of the problem on “piecewise uniform” rectangular meshes which are locally condensing with respect to both
$x$ and
$t$. Using stencils with nonorthogonal (in
$x$ and
$t$) arms in the nearest neighbourhood of the set
$\gamma^*$ and meshes condensing, along
$x$, in the neighbourhood of
$\gamma^*$, we construct schemes that converge euniformly with the rate
$\mathscr O(N^{-k}\ln^kM+N_0^{-1})$,
$k=1,2$.
Received: 12.04.2002
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