The figure shows sets of discrete points with indices {i-1,i,i+1}. The formulation of the first derivative by finite differences depends on the sign of a. Upwinding schemes consider the ”flow”-direction of the information and only take the cor- responding points (marked with dots) into account. The left part of the figure shows the points used for a first order upwinding scheme for a > 0. The right part of the figure shows the same for a < 0. The discretised version of the above equation using a first order upwind scheme is given by:

If we apply this scheme to a function (some sort of Gauss peak) with homogenous Dirichlet boundary conditions (we chose the size of the simulation box in a way to justify the boundary conditions) we would expect that the peak is propagated as a function of time without changing its shape. The following figure shows the result at different times ...

Obviously the peak broadens when propagating through the time - is diffusion the reason for this ? And if so, where does the diffusion come from ?