A finite-difference approximation of a two-layer system for growing sandpiles

We study from a numerical point of view a system of partial differential equations recently proposed in [K. P. Hadeler and C. Kuttler, Granular Matter, 2 (1999), pp. 9-18] to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded table. In such a system, an eikonal equation for the standing layer of the pile is coupled to an advection equation for the rolling layer. We analyze a suitable explicit finite difference scheme in one dimension and discuss its properties. A comparison is made with respect to the variational approach of [L. Prigozhin, European J. Appl. Math., 7 (1996), pp. 225-235], particularly in terms of steady-state solutions. Since the effective equilibrium configuration reached by the growing process is not completely clear for this model, we show experiments in one and two dimensions, which characterize the steady-state solutions for different source terms. © 2006 Society for Industrial and Applied Mathematics.