  Numerical climate- and weather-prediction requires the fast solution of the
equations of fluid dynamics. Discontinuous Galerkin (DG) discretisations have
several advantageous properties. They can be used for arbitrary domains and
support a structured data layout, which is important on modern chip
architectures. For smooth solutions, higher order approximations can be
particularly efficient since errors decrease exponentially in the polynomial
degree. Due to the wide separation of timescales in atmospheric dynamics,
semi-implicit time integrators are highly efficient, since the implicit
treatment of fast waves avoids tight constraints on the time step size, and can
therefore improve overall efficiency. However, if implicit-explicit (IMEX)
integrators are used, a large linear system of equations has to be solved in
every time step. A particular problem for DG discretisations of
velocity-pressure systems is that the normal Schur-complement reduction to an
elliptic system for the pressure is not possible since the numerical fluxes
introduce artificial diffusion terms. For the shallow water equations, which
form an important model system, hybridised DG methods have been shown to
overcome this issue. However, no attention has been paid to the efficient
solution of the resulting linear system of equations. In this paper we address
this issue and show that the elliptic system for the flux unknowns can be
solved efficiently with a non-nested multigrid algorithm. The method is
implemented in the Firedrake library and we demonstrate the excellent
performance of the algorithm both for an idealised stationary flow problem in a
flat domain and for non-stationary setups in spherical geometry from the
Williamson et al. testsuite. In the latter case the performance of our bespoke
multigrid preconditioner (although itself not highly optimised) is comparable
to that of a highly optimised direct solver.
