Internal layer intersecting the boundary of a domain in a singular advection-diffusion equation
Résumé
We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection-diffusion equation y ε t +M (x, t)y ε x −εy ε xx = 0, (x, t) ∈ (0, 1)×(0, T), supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O(ε) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O(ε 1/2) 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 P ε satisfying y ε − P ε L ∞ (0,T ;L 2 (0,1)) = O(ε 1/2). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.
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