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Even better Implicit Runge--Kutta methods
There's a few things we could do to tidy up their implementation even further. Most of these are primarily considered in the FIRK case, so it might need some thinking to generalise them to the general implicit case.
- Iterate closer to zero. See (8.1) of Hairer & Wanner.
- Smarter stage-value predictors using interpolation. See (8.5') of Hairer & Wanner.
- Transform linear problems into Hessenberg form. See (8.15) of Hairer & Wanner.
- Stiff error estimators: adjust the error estimators of implicit solvers to depend on the implicit problems being solved. See (8.19) and (8.20) of Hairer & Wanner.