  Most research on preconditioners for time-dependent PDEs has focused on
implicit multi-step or diagonally-implicit multi-stage temporal
discretizations. In this paper, we consider monolithic multigrid
preconditioners for fully-implicit multi-stage Runge-Kutta (RK) time
integration methods. These temporal discretizations have very attractive
accuracy and stability properties, but they couple the spatial degrees of
freedom across multiple time levels, requiring the solution of very large
linear systems. We extend the classical Vanka relaxation scheme to implicit RK
discretizations of saddle point problems. We present numerical results for the
incompressible Stokes, Navier-Stokes, and resistive magnetohydrodynamics
equations, in two and three dimensions, confirming that these relaxation
schemes lead to robust and scalable monolithic multigrid methods for a
challenging range of incompressible fluid-flow models.
