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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.

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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