Asymptotic solution of a nonlinear advection-diffusion equation
Romain de Loubens, T.S.. S. Ramakrishnan · Quarterly of Applied Mathematics · 2011
We carry out an asymptotic analysis as t → ∞ t\rightarrow \infty for the nonlinear advection-diffusion equation, ∂ t u = 2 α u ∂ x u + ∂ x ( u ∂ x u ) \partial _t u = 2\alpha u \partial _x u + \partial _x(u \partial _x u) , where α \alpha is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with α \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 O ( t ) \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 O ( 1 / t ) \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 O ( 1 ) \mathcal {O}(1) , as well as a composite solution correct to O ( 1 / t ) \mathcal {O}(1/t)