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

Numerical climate- and weather-prediction models require 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 particularly 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 by using 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 well-known testsuite in [Williamson et al. (1992) JCP, 102(1), pp.211-224]. 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.