Asymptotic solution of a nonlinear advection-diffusion equation

We carry out an asymptotic analysis as $t\rightarrow \infty$ for the nonlinear advection-diffusion equation, $\partial _t u = 2\alpha u \partial _x u + \partial _x(u \partial _x u)$, where $\alpha$ is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with $\alpha$ having a specific physical significance related to the bed inclination. For compactly supported initial data, the solution is characterized by two moving boundaries propagating with finite speed and spanning a distance of $\mathcal {O}(\sqrt {t})$. We construct an exact outer solution to the PDE that satisfies the right boundary condition. The vanishing condition at the left boundary is enforced by introducing a moving boundary layer, for which we obtain a closed-form expression. The leading-order composite solution is uniformly correct to $\mathcal {O}(1/\sqrt {t})$. A higher-order correction to the inner and the composite solutions is also derived analytically. As a result, we obtain late-time asymptotic expansions for the two moving boundaries, correct to $\mathcal {O}(1)$, as well as a composite solution correct to $\mathcal {O}(1/t)$. The findings of this paper are illustrated and verified by numerical computations.