We study a system of coupled convection-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them which give rise to boundary layers at either boundary. An upwind finite difference scheme on arbitrary meshes is used to solve the system numerically. A general error estimate is derived that allows to immediately conclude robust convergence – w.r.t. the perturbation parameters – for certain layer-adapted meshes, thus improving and generalising previous results . We present the results of numerical experiments to illustrate our theoretical findings.
AMS Subject Classifications
65L10 65L12 65L60
Convection-diffusion singular perturbation layer-adapted mesh systems of odes derivative bounds
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Andreev, V. B., Kopteva, N. V. 1998On the convergence, uniform with respect to a small parameter, of monotone three-point finite-difference approximationsDiffer. Equations34921929zbMATHMathSciNetGoogle Scholar
Andreev, V. B. 2001The Green function and a priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemesDiffer. Equations37923933zbMATHCrossRefGoogle Scholar
Bakhvalov, N. S. 1969Towards optimization of methods for solving boundary value problems in the presence of boundary layersZh. Vychisl. Mat. i Mat. Fiz.9841859(in Russian)zbMATHGoogle Scholar
Cen, Z. 2005Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equationsJ. Sys. Sci. Complexity18498510zbMATHMathSciNetGoogle Scholar
Kellogg, R. B., Tsan, A. 1978Analysis of some difference approximations for a singular perturbation problem without turning pointsMath. Comput.3210251039zbMATHCrossRefMathSciNetGoogle Scholar
Linß, T.: Layer-adapted meshes for convection-diffusion problems. Habilitation thesis, Technische Universität Dresden 2006.Google Scholar
Linß, T., Madden, N.: Layer-adapted meshes for a system of coupled singularly perturbed reaction-diffusion problem. (2007, in preparation).Google Scholar
Madden, N., Stynes, M. 2003A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problemsIMA J. Numer. Anal.23627644zbMATHCrossRefMathSciNetGoogle Scholar
Miller, J. J. H., O'Riordan, E., Shishkin, G. I. 1996Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensionsWorld ScientificSingaporezbMATHGoogle Scholar
Morton, K. W. 1996Numerical solution of convection-diffusion problems. Applied Mathematics and Mathematical Computation, vol. 12Chapman & HallLondonGoogle Scholar