An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers

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• 作者：C. Clavero^a ; ^{clavero@unizar.es} ; J.L. Gracia^a ; ^{jlgracia@unizar.es} ; G.I. Shishkin^b ; ^{shishkin@imm.uran.ru} ; L.P. Shishkina^b ; ^{lida@convex.ru}
• 关键词：1D parabolic singularly perturbed problems ; Two parameters ; Discontinuous source ; Interior and boundary layers ; Uniform convergence
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：July 2017
• 年：2017
• 卷：318
• 期：Complete
• 页码：634-645
• 全文大小：483 K
• 卷排序：318

文摘

In this paper we consider a 1D parabolic singularly perturbed reaction-convection–diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter ε2ε2) and the convection term (multiplied by the parameter μμ) in the differential equation (ε∈(0,1]ε∈(0,1], μ∈[0,1]μ∈[0,1], μ≤εμ≤ε). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in εε and μμ, having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed.