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

Abstract

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