Late-time asymptotic solution of nonlinear advection-diffusion equations with equal exponents

We derive the late-time asymptotic solution of a nonlinear advection-diffusion equation, $u_t = [\alpha /(q-1)](u^q)_x + (1/q) (u^q)_{xx}$, where $\alpha \ne 0$ and $q > 2$. The equation is a more general form of the purely quadratic nonlinearity for advection and diffusion considered previously. For initial conditions with compact support, the solution has left and right moving boundaries, the distance between which is the width of the “plume”. We show the width to grow as $t^{1/q}$, with a constant correction term. The outer solution is dominated by the nonlinear advective term, the leading-order solution of which is shown to satisfy the partial differential equation and the right boundary condition exactly, but with a $t$-dependent shifted argument. To satisfy the left boundary of vanishing plume thickness, a boundary layer is introduced, for which the inner solution may be obtained up to second order, again by using a shifted coordinate with respect to the wetting front. A leading-order composite solution for $u$, uniformly correct to $O(1/t^{1/q})$, is obtained. The first and second-order terms are correct to $O((1/t^{2/q})\ln t)$ and $O(1/t^{2/q})$ respectively. The composite second-order correction involves an arbitrary constant, implying its dependence on an unknown initial condition. Numerical results that agree with the analytical solutions are given along with an expression for the unknown constant computed with an impulse initial data.