Detecting two-dimensional fingering patterns in a non-equilibrium pde model via adaptive moving meshes

This article discusses an adaptive mesh method applied to a bifurcation problem in a non-equilibrium Richard’s equation from hydrology. The extension of this PDE model for the water saturation S, to take into account additional dynamic memory effects gives rise to an extra third-order mixed space-time derivative term in the PDE. The one-space dimensional case predicts the formation of steep non-monotone waves depending on the non-equilibrium parameter. In two space dimensions, this parameter and the frequency in a small perturbation term, predict that the waves may become unstable, thereby initiating so-called gravity-driven fingers. To detect the steep solutions of the time-dependent PDE model, we have used a sophisticated adaptive moving mesh method based on a scaled monitor function.