Adaptive stepper

[danvau98]

Additional IMEX scheme with an embedded scheme to allow for adaptive stepping.

[danvau98]

I want to find some lower-order embedded methods that would increase the speed of these solvers. The Kennedy and Carpenter schemes are 4th and 5th-order; ideally, I should find a 3rd and 2nd-order scheme to add.

[danvau98]

Hi @bpbrown and @kburns, any suggestions for how this can be improved?

[brownbaerchen]

This is interesting. I have a few questions:

• Did you check that you get the desired order of accuracy for both methods? I understand that you cannot just sum because of algebraic constraints. But if you solve, you are doing something different. If you get the correct order, is it by chance or is it expected?
• Why did you comment out the above line? I would expect that it is needed in general.
• How many more factorisations do you need with this scheme? The RK methods so far implemented in Dedalus are all singly diagonally implicit, such that you need only one factorisation per step. But I assume $\hat{b}s \neq H{ss}$, such that you need one more factorisation. If the step size is kept constant, do you need to factorize twice again in the next step or is there a cache?