Dynamic discrete model for the granular matter formation process

In this paper we introduce 1-$D$ and 2-$D$ discrete models for the dynamic granular matter formation process in the form of a system of difference equations. This approach allows us to differentiate between the influx of the rolling layer coming down from different directions to the corner points of the standing layer. Such points are difficult to adequately describe by means of pde's and their straightforward numerical approximations, typically ``ignoring'' the system's behavior on the sets of zero measure. However, these points are critical for understanding the dynamic formation process when the standing layer is created by the moving front of the rolling matter or when the latter is filling a cavity and/or stops rolling. The existence of distributed (infinite dimensional) limit solutions to our discrete models as the size of mesh tends to zero is also discussed. We illustrate our findings by numerical examples which use our model as a direct algorithm.