Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation

G. Shishkin ()
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G. Shishkin: Russian Academy of Sciences, Institute of Mathematics and Mechanics

A chapter in Numerical Mathematics and Advanced Applications 2011, 2013, pp 293-301 from Springer

Abstract: Abstract For a model Dirichlet problem to a singularly perturbed ordinary differential convection-diffusion equation, we discuss a “standard” approach to the construction of difference schemes that use standard grid approximations on uniform grids, the step-size of which is chosen sufficiently small for small values of a perturbation parameter $$\varepsilon $$ , $$\varepsilon \in (0,1]$$ . It is shown that such a scheme, under its convergence in the maximum norm theoretically proved, is not $$\varepsilon $$ -uniformly stable to perturbations in the data of the discrete problem. When perturbations take place and the parameter $$\varepsilon $$ decreases, the actual accuracy of the computed solutions may deteriorate up to a full accuracy loss for sufficiently small values of $$\varepsilon $$ , namely, under the condition $$t = \mathcal{O}(\ln {\varepsilon }^{-1})$$ , where t is the number of computer word digits.

Keywords: Classical Difference Schemes; Piecewise Uniform Grid; Model Dirichlet Problem; Majorant Function Technique; Numerical Grid Methods (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-33134-3_32

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DOI: 10.1007/978-3-642-33134-3_32

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