OpenFOAM offers ODE solvers for non-stiff systems including: fifth-order Cash-Karp Runge-Kutta with error estimation and adaptive time step control (Cash, J.R. and Karp, A.H., ACM transactions on mathematical software, 16, pp. 201-222, 1990).
OpenFOAM offers ODE solvers for stiff systems including: the fourth-order semi-implicit Runge-Kutta scheme of Kaps, Rentrop and Rosenbrock with error estimation and adaptive time step control (Kaps, P. and Rentrop, P. Numerische Mathematik, 33, pp. 55-68, 1979); and, the semi-emplicit Bulirsch-Stoer method of Bader and Deuflhard (Bader, G. and Deuflhard, P., Numerische Mathematik, 41, pp. 373-398, 1983).