Code Release: 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, hybridized 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 show how 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 superior performance of the algorithm for a representative model problem.