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

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.