  This paper is concerned with the development and testing of advanced
time-stepping methods suited for the integration of time-accurate, real-world
applications of computational fluid dynamics (CFD). The performance of several
time discretization methods is studied numerically with regards to
computational efficiency, order of accuracy, and stability, as well as the
ability to treat effectively stiff problems. We consider matrix-free
implementations, a popular approach for time-stepping methods applied to large
CFD applications due to its adherence to scalable matrix-vector operations and
a small memory footprint. We compare explicit methods with matrix-free
implementations of implicit, linearly-implicit, as well as Rosenbrock-Krylov
methods. We show that Rosenbrock-Krylov methods are competitive with existing
techniques excelling for a number of problem types and settings.
