For the solution of many PDE, implicit high-order time integration schemes are required. I am specifically interested in schemes that do not require a constant time step.
I am well acquainted with BDF time integration schemes and I am aware of their (sometimes harsh) limitations in terms of stability but also in terms of time step change.
I have seen SDIRK schemes mentioned extensively in the literature. From what I gather, they are multi-step implicit methods that require the solution of multiple regularly sized linear systems (i.e not bigger than what implicit Euler would give) at each iterations. However, the majority of the articles I have seen are more a discussion on the stability rather than implementation. Furthermore, the majority focus on order 4 and higher, but 2nd or 3rd order would be closer to my needs.
Is there a simple reference in the literature that highlights the numerical scheme to implement and what are the limitations in terms of stability and time step change? I have not been able to find something that was sufficiently simple and straight to the point.