A Robust Numerical Method for a Singularly Perturbed Parabolic Convection-Diffusion Problem with a Degenerating Convective Term and a Discontinuous Right-Hand Side

C. Clavero (), J. L. Gracia (), G. I. Shishkin () and L. P. Shishkina ()
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C. Clavero: University of Zaragoza, School of Engineering and Architecture
J. L. Gracia: University of Zaragoza, School of Engineering and Architecture
G. I. Shishkin: Ural Branch of Russian Academy of Sciences, Institute of Mathematics and Mechanics
L. P. Shishkina: Ural Branch of Russian Academy of Sciences, Institute of Mathematics and Mechanics

A chapter in Numerical Mathematics and Advanced Applications 2011, 2013, pp 257-265 from Springer

Abstract: Abstract In this paper we consider the efficient numerical approximation of a singularly perturbed parabolic convection-diffusion problem having a convective term which degenerates inside the domain, in the case that the right-hand side of the differential equation is discontinuous on the degeneration line. For small values of the diffusion parameter $${\varepsilon }^{2}$$ ( $$\varepsilon \in (0,1]$$ ), in general, the exact solution has an interior layer in a neighborhood of the degeneration line. We construct a classical finite difference scheme combining the implicit Euler method in time, defined on a uniform mesh, and the first order upwind scheme in space, defined on a piecewise-uniform grid condensing in a neighborhood of the interior layer. Then, the method is an $$\varepsilon $$ -uniformly convergent scheme of first order in time and almost first order in space. We show the numerical results for a test problem, confirming in practice the theoretical results.

Keywords: Discontinuous Right-hand Side; Interior Layers; Finite Difference Scheme; Piecewise Uniform Mesh; Degeneration Line (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_28

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DOI: 10.1007/978-3-642-33134-3_28

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