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Numerical climate- and weather-prediction requires the fast solution of theequations of fluid dynamics. Discontinuous Galerkin (DG) discretisations haveseveral advantageous properties. They can be used for arbitrary domains andsupport a structured data layout, which is important on modern chiparchitectures. For smooth solutions, higher order approximations can beparticularly efficient since errors decrease exponentially in the polynomialdegree. Due to the wide separation of timescales in atmospheric dynamics,semi-implicit time integrators are highly efficient, since the implicittreatment of fast waves avoids tight constraints on the time step size, and cantherefore improve overall efficiency. However, if implicit-explicit (IMEX)integrators are used, a large linear system of equations has to be solved inevery time step. A particular problem for DG discretisations ofvelocity-pressure systems is that the normal Schur-complement reduction to anelliptic system for the pressure is not possible since the numerical fluxesintroduce artificial diffusion terms. For the shallow water equations, whichform an important model system, hybridised DG methods have been shown toovercome this issue. However, no attention has been paid to the efficientsolution of the resulting linear system of equations. In this paper we addressthis issue and show that the elliptic system for the flux unknowns can besolved efficiently with a non-nested multigrid algorithm. The method isimplemented in the Firedrake library and we demonstrate the excellentperformance of the algorithm both for an idealised stationary flow problem in aflat domain and for non-stationary setups in spherical geometry from theWilliamson et al. testsuite. In the latter case the performance of our bespokemultigrid preconditioner (although itself not highly optimised) is comparableto that of a highly optimised direct solver.

Multigrid preconditioners for the hybridized Discontinuous Galerkin discretisation of the shallow water equations