Imagine we have an empty table 
and an unlimited number of balls are accumulated at the right side (the gradient is 100%).
How can we numerically calculate/simulate each and every move from this empty state to fully filled state?
For diffusion or heat equation, we solve the partial differential equation with these boundary conditions. Instead of the differential equation, I wish to find a numerical approach to calculate the distribution of balls into the table at any step.
Furthermore, I appreciate some references of the numerical methods for solving this specific model.