Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model

Abstract

Due to the wide separation of time-scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves, while not suffering from severe restrictions on the time-step size. To propagate the state of the atmosphere forward in time, a nonlinear equation for the prognostic variables has to be solved at every time step. Since the nonlinearity is typically weak, this is done with a small number of Newton or Picard iterations, which in turn require the efficient solution of a large system of linear equations with (Formula presented.) unknowns. This linear solve is often the computationally most costly part of the model. In this article an efficient linear solver for the LFRic next-generation model currently being developed by the Met Office is described. The model uses an advanced mimetic finite element discretisation which makes the construction of efficient solvers challenging as compared to models using standard finite-difference and finite-volume methods. The linear solver hinges on a bespoke multigrid preconditioner of the Schur-complement system for the pressure correction. By comparing it to Krylov subspace methods, the superior performance and robustness of the multigrid algorithm is demonstrated for standard test cases and realistic model set-ups. In production mode, the model will have to run in parallel on hundreds of thousands of processing elements. As confirmed by numerical experiments, one particular advantage of the multigrid solver is its excellent parallel scalability due to its avoidance of expensive global reduction operations.