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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 (Robin boundary), 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? (edit: I have no more oscillations)

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    $\begingroup$ Can you post what your discretization with the Theta scheme looks like? A stability analysis will tell you if it is important. $\endgroup$
    – nluigi
    Commented Nov 30, 2015 at 10:49
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    $\begingroup$ $\frac{c^{n+1}-c^n}{\tau} = \Theta (d \nabla^2 c^{n+1} - q \nabla c^{n+1}) + (1- \Theta)(d \nabla^2 c^{n} - q \nabla c^{n})$ q is the velocity and d the diffusion coefficient. I applied an upwind scheme for $\nabla^2 c^{n+1}$ and $\nabla^2 c^{n}$ (forward difference for negative q and vice versa). $\nabla^2$ is approximated with the 4 point stencil. $\endgroup$
    – Paulinchen2
    Commented Nov 30, 2015 at 11:23
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    $\begingroup$ This scheme is only unconditionally stable for $\theta\ge 0.5$ as far as the time stepping method is concerned. Of course, you will still have to stabilize the transport term. $\endgroup$
    – Wolfgang Bangerth
    Commented Nov 30, 2015 at 15:06

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