Just to be on the clear side, I was only quoting part of their conclusion. They do also make some remarks in favor of Julia. But perhaps the rebuttal of that paper can go to a new thread or the Julia Discourse.
To return to the original question, I have personally used the following ODE solvers:
- dop853 (thanks to @jacobwilliams)
- ddeabm (also thanks to @jacobwilliams)
- RKC: An explicit solver for parabolic PDEs (available from netlib)
- stiff3
In a distant past I also used LSODE from ODEPACK which gave good results, but the API is rather cumbersome. Another problem with many of the Fortran ODE solvers is they are not thread-safe.