As $\varepsilon$ goes to zero, the unique solution of the scalar advection-diffusion equation $y^{ \varepsilon}_t-\varepsilon y^{ \varepsilon}_{ xx} + M y^{ \varepsilon}_x=0$, $(x,t)\in (0,1)\times (0,T)$ submitted to Dirichlet boundary conditions exhibits a boundary layer of size $\mathcal{ O}(\varepsilon)$ and an internal layer of size $\mathcal{ O}(\sqrt{ \varepsilon})$. If the time $T$ is large enough, these thin layers where the solution $y^{ \varepsilon}$ displays rapid variations intersect and interact each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation $\widetilde{ P}^\varepsilon$ of the solution $y^\varepsilon$ satisfying $\Vert y^{ \varepsilon}-\widetilde{ P}^\varepsilon\Vert_{ L^\infty(0,T; L^2(0,1))}=\mathcal{ O}(\varepsilon^{ 3/2})$ and $\Vert y^{ \varepsilon}-\widetilde{ P}^\varepsilon\Vert_{ L^2(0,T; H^1(0,1))}=\mathcal{ O}(\varepsilon)$, for all $\varepsilon$ small enough.
Categories: math.AP