I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of $10^{-15}$). The boundary conditions are: open boundary on two sides, and Dirichlet boundary $u=0$ on the other two.

I have some questions

• Is the CFL-Number of any importance when solving the Convection Diffusion Equation in 2D using the $\theta$ scheme and Finite Differences?

• How does the diffusion coefficient factor into the CFL-Condition?

• I know the implicit case is supposed to be stable for all time steps and step sizes, but I get ugly oscillations. Is it only important for the explicit case?