Why cant I get a stable Cranck Nicolson discretization?

Hi all,

For a project we try to design a reactor with a diffusion-convection-reaction mechanism before it reaches a steady state. Now, we are using the Cranck Nicoloson discretization for this reaction. However this does not seem to work properly. We get an unstable system, because we have a low diffusion coefficient. We have been playing around, but we cannot find how we can make this system more stable.

Attached is the code we are using right now.

Do you maybe have suggestions on how to make the discretization more stable? Many thanks!

edit: the initial mass balance we are trying to solve is dx/dt = D*d^2x/da^2+v_av*void*dx/da-k*void*x

We use a Dirichet boundary condition so that x=0 for t = 0 and a = 0.

[Torsten]

Could you include the equation you are trying to solve together with initial and boundary conditions ?

[Jort Puiman]

Hi Torsten! I edited the question so that it contains our initial mass balance and dirichet condition

[Bjorn Gustavsson]

@Jort Puiman: were our suggestions sufficient to solve your issue?

[Jort Puiman]

Unfortunately, our model was too sensitive, and our lecturer made us switch to the method of lines discretization.

The timesteps were indeed problematic, but smaller timesteps did not fully fix the problem, and the model remained unstable after a longer time.

Answer 1

[Bjorn Gustavsson]

(caveat: I didn't run your code or analyze it, but speak from my experience with CN) Most likely you take too long steps in time. Even for CN-methods one has to obey the CFL-condition when selecting the time-steps. You might have to read up on the Courant-Friedrichs-Levy condition: wiki-on-CFL-condition.

That was my experience, didn't think I needed to bother because CN is stable, but first solutions were not good, when the CFL-condition on dt was adhered to all was good.

HTH

Answer 2

[Torsten]

We use a Dirichet boundary condition so that x=0 for t = 0 and a = 0.

And which conditions at a = a_end ?

Try "pdepe".

If the convection term v_av*void*dx/da is too strong compared to the diffusion term, discretize dx/da using an upwind method, not central differences.