
This article is cited in 1 scientific paper (total in 1 paper)
Improved approximations of the solution and derivatives to a singularly perturbed reactiondiffusion equation based on the solution decomposition method
G. I. Shishkin^{}, L. P. Shishkina^{} ^{} Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16,
Yekaterinburg, 620990 Russia
Abstract:
In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction–diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first and secondorder derivatives converge in the maximum norm uniformly with respect to
a perturbation parameter $\varepsilon\in(0,1]$; the normalized derivatives are $\varepsilon$uniformly bounded. The key idea of this approach to the construction of $\varepsilon$uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first and secondorder derivatives converge $\varepsilon$uniformly at the rate of $O(N^{2}\ln^2N)$, where $N+1$ is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge $\varepsilon$uniformly in the maximum norm at the same rate of $O(N^{4}\ln^4N)$.
Key words:
singularly perturbed boundary value problem, ordinary differential reaction–diffusion equation, decomposition of grid solution, asymptotic construction technique, finite difference scheme of the solution decomposition method, uniform grids, $\varepsilon$uniform convergence, maximum norm, the Richardson technique, improved scheme of the solution decomposition method, improved approximation of derivatives.
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Computational Mathematics and Mathematical Physics, 2011, 51:6, 1020–1049
Bibliographic databases:
UDC:
519.633 Received: 15.11.2010
Citation:
G. I. Shishkin, L. P. Shishkina, “Improved approximations of the solution and derivatives to a singularly perturbed reactiondiffusion equation based on the solution decomposition method”, Zh. Vychisl. Mat. Mat. Fiz., 51:6 (2011), 1091–1120; Comput. Math. Math. Phys., 51:6 (2011), 1020–1049
Citation in format AMSBIB
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\pages 10911120
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\pages 10201049
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http://mi.mathnet.ru/eng/zvmmf9466 http://mi.mathnet.ru/eng/zvmmf/v51/i6/p1091
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G. I. Shishkin, L. P. Shishkina, “A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reactiondiffusion equation”, Comput. Math. Math. Phys., 55:3 (2015), 386–409

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