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The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition. (Russian, English) Zbl 1199.65284
Comput. Math., Math. Phys. 49, No. 8, 1348-1368 (2009); translation from Zh. Vychisl. Mat. Mat. Fiz. 49, No. 8, 1416-1436 (2009).
Summary: 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 $$O(N^{-2}\ln^2N_0^{-1} )$$, 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 $$O(N^{-3} + N_0^{-2})$$ 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.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 35B25 Singular perturbations in context of PDEs
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