Asymptotic analysis of an advection‐diffusion equation involving interacting boundary and internal layers

Abstract

As ε goes to zero, the unique solution of the scalar advection-diffusion equation y ε t −εy ε xx +M y ε x = 0, (x, t) ∈ (0, 1) × (0, T ) submitted to Dirichlet boundary conditions exhibits a boundary layer of size O(ε) and an internal layer of size O( √ ε). If the time T is large enough, these thin layers where the solution y ε displays rapid variations intersect and interact each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation P ε of the solution y ε satisfying y ε − P ε L ∞ (0,T ;L 2 (0,1)) = O(ε 3/2 ) and y ε − P ε L 2 (0,T ;H 1 (0,1)) = O(ε), for all ε small enough.