A Solution to a Homework Problem

In last week physic's friday blog post The diffusion in convection I suggested to do a little homework - performing the same analysis with the first order unwinding scheme as we have done with the central difference scheme. If you went through these steps you should end up (in the a>0 case) with

Using

and defining

one ends up with

.

Therefore, the scheme is stable for C<=1 and of order one. We can identify the second order derivative, introducing the numerical diffusion we observed, when propagating a simple Gaussian like peak with our evolution equation (see: A convection toy problem or Can you find the diffusion).

How can this diffusion be reduced ? - With higher order unwinding schemes additional terms are introduced that are responsible for the reduction. 

Solution of the first order evolution equation at t = 0.0 (blue) and t = 1.0 using the first order upwinding scheme (red) and the Lax-Wendroff scheme (a higher order scheme) (yellow). Obviously, the diffusive behavior of the first order upwinding scheme is reduced by the anti-diffusion term introduced by the Lax-Wendroff scheme.