DAE Interactive Adaptive Step-Size Solver

Hi, I have a DAE model with explicit algebraic equations (I would like to highlight this since I've already see the issue about DAE with implicit algebraic equations). I just implemented a simple step-by-step solver instead of using the diffeqsolve to handle algebraic equations. However, I found a paricular DAE that is not solvable using classical Dopri5 or Tsit5, unless using very small step size, hence I decided to use an adaptive stepsize controller (the PIDController) and the solver Kvaerno5.

To be more clear this is a system I would like to deal with, like any biological systems.

dx/dt = f(x(t),args,t)
args_1 = g_1(...)
args_2 = g_2(...)
...
args_N = g_N(...)

Having said that, I'm actually having trouble implementing the same step-by-step using the PIDController and the SaveAt. Are there any suggestions?

Thank you for the answer.

I'm in particular interest for this project and Equinox too, they are awesome!!

Start by reproducing the reported DAE behavior with the step-by-step solver, PIDController, SaveAt, and Kvaerno5 mentioned in the issue. Compare that setup with diffeqsolve and define the expected adaptive-step behavior for explicit algebraic equations before determining what a complete solution would require.