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

As e goes to zero, the unique solution of the scalar advection-diffusion equation y e t −ey e xx +M y e x = 0, (x, t) ∈ (0, 1) × (0, T) submitted to Dirichlet boundary conditions exhibits a boundary layer of size O(e) and an internal layer of size O(√ e). If the time T is large enough, these thin layers where the solution y e 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 e of the solution y e satisfying y e − P e L ∞ (0,T ;L 2 (0,1)) = O(e 3/2) and y e − P e L 2 (0,T ;H 1 (0,1)) = O(e), for all e small enough.