  We conduct a comparative study of the Jacobian-free linearly implicit
Rosenbrock-Wanner (ROW) methods, the explicit first stage, singly diagonally
implicit Runge-Kutta (ESDIRK) methods, and the second-order backward
differentiation formula (BDF2) for unsteady flow simulation using spatially
high-order flux reconstruction/correction procedure via reconstruction (FR/CPR)
formulations. The pseudo-transient continuation is employed to solve the
nonlinear systems resulting from the temporal discretizations with ESDIRK and
BDF2. A Jacobian-free implementation of the restarted generalized minimal
residual method (GMRES) solver is employed with a low storage element-Jacobi
preconditioner to solve linear systems, including those in linearly implicit
ROW methods and those from linearization of the nonlinear systems in ESDIRK and
BDF2 methods. We observe that all ROW and ESDIRK schemes (from second order to
fourth order) are more computationally efficient than BDF2, and ROW methods can
potentially be more efficient than ESDIRK methods. However, the convergence
tolerance of the GMRES solver for ROW methods needs to be sufficiently tight to
preserve the nominal order of accuracy. In general, ESDIRK methods allow a
larger physical time step size for unsteady flow simulation than ROW methods
do.
