I'm currently working on a computational model of the way sand acts when being pushed around, say, by a bulldozer. Specifically, I'm trying to determine the dynamic equations which govern the way that sand will move under gravity from an arbitrary intial configuration (say, a narrow tower) toward a position of equilibrium (the stationary pile formed after the tower collapses). That way I can apply the material shifts from the dozer while applying these equations at regular timesteps to achieve what is hopefully a physically representative model.
I'm modelling the sand using a heightmap (a 2D grid where each cell holds the corresponding height of the sand at that point), and I'm currently treating height like "concentration" and using particle diffusion equations in 2D:
$$\frac{\partial C(x, t)}{\partial t}=-D\cdot\nabla^2C(x, t)$$
The discrete-time equivalent is then generating a Gaussian diffusion kernel at each timestep that I convolve with the heightmap to get the next state, but this model tends towards a completely flat equilibrium. I was wondering what the appropriate equations would be to use for this model? Even a pointer towards some relevant resources would be hugely useful. I know that DEM is an alternative and for now I'm trying to avoid it.
Thanks!
Edit: From research I understand that the dynamics of granular materials are very complex - I was wondering if there was a simplified modelling approach that necessarily makes a few basic assumptions but enables the kind of modelling that I'm after.
Edit: I used @KGM's answer and it worked great. Computing the necessary Lipschitz function isn't trivial but can be optimised quite well afterwards.