Asymptotic solution of a nonlinear advection-diffusion equation

Abstract

We carry out an asymptotic analysis as t → ∞ for the nonlinear advection-diffusion equation, ∂ t u = 2αu∂ x u + ∂ x (u∂ x u), where α is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with α 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). 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). 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), as well as a composite solution correct to O(1/t). The findings of this paper are illustrated and verified by numerical computations.