The diffusion in convection

In last weeks blog post "A convection toy problem or can you find the diffusion" I showed the picture of a function propagating through the time. Although the analytical solution of the problem suggests that the shape of the peak remains stable as a function of time, the peak obviously broadens indicating that (numerical) diffusion occurs.

If we go back one step and use a central difference scheme for the discretization of the first order spatial derivative we obtain the following result

We clearly see that during propagation the peak does not only change its shape but that scheme becomes even unstable. If we can analyse this behaviour perhaps we get an answer to our first problem, namely how numerical diffusion is introduced into the stable propagation when using an upwind scheme.

We use a Taylor expansion for the following equation

and obtain

Using the last but one equation with the last one we obtain

Numerical diffusion is introduced due to the discretisation scheme used — the negative diffusion coefficient in the leading error term makes the scheme unconditionally unstable. As a homework I suggest to perform this kind of analysis to the unwinding scheme (one sided difference) to answer the question where the diffusion comes from. I will post the solution next week.