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Article type: Research Article
Authors: Amirat, Youcef | Münch, Arnaud; *
Affiliations: Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
Correspondence: [*] Corresponding author. E-mail: arnaud.munch@uca.fr.
Abstract: We perform an asymptotic analysis with respect to the parameter ε>0 of the solution of the scalar advection–diffusion equation ytε+M(x,t)yxε−εyxxε=0, (x,t)∈(0,1)×(0,T), supplemented with Dirichlet boundary conditions. For small values of ε, the solution yε exhibits a boundary layer of size O(ε) in the neighborhood of x=1 (assuming M>0) and an internal layer of size O(ε1/2) in the neighborhood of the characteristic starting from the point (0,0). Assuming that these layers interact each other after a finite time T>0 and using the method of matched asymptotic expansions, we construct an explicit approximation Pε satisfying ‖yε−Pε‖L∞(0,T;L2(0,1))=O(ε1/2). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.
Keywords: Asymptotic analysis, singular perturbation, internal and boundary layers, Sobolev estimates
DOI: 10.3233/ASY-231836
Journal: Asymptotic Analysis, vol. 134, no. 3-4, pp. 297-343, 2023
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