Jacobian-dependent vs Jacobian-free discretizations for nonlinear differential problems

The paper provides a comparison between two relevant classes of numerical discretizations for stiff and nonstiff problems. Such a comparison regards linearly implicit Jacobian-dependent Runge–Kutta methods and fully implicit Runge–Kutta methods based on Gauss–Legendre nodes, both A-stable. We show that Jacobian-dependent discretizations are more efficient than Jacobian-free fully implicit methods, as visible in the numerical evidence.