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

Abstract

Due to the wide separation of timescales 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 (10 6 − 10 9) 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 setups. 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.

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

“…We use an 'implicit Richardson' preconditioner P as reference to allow for a qualitative comparison between conventional and machine learning approaches (see supporting information), that is based on performing implicit Richardson iterations in zonal direction to diminish the effects of grid-convergence near the poles. This approach is equivalent to the well-known treatment of the vertical dimension with a tridiagonal solver, see (Müller & Scheichl, 2014;Maynard et al, 2020). The preconditioner has been successfully tested for shallow-water test-cases from the Williamson test-suite (Williamson et al, 1992) and allowed for solver speed-ups of factor 3 to 10 (not shown here).…”

“…However, for implicit models the linear solver is responsible for the majority of the computational cost. This paper is focussing on the class of Krylov sub-space methods but the approach presented should also be relevant for multigrid-based methods, see (Müller & Scheichl, 2014;Maynard et al, 2020) for recent publications on linear solvers.…”

“…A preconditioner directly inverts specific parts of the linear problem in order to significantly reduce the number of solver iterations. However, deriving efficient preconditioners is a difficult exercise requiring substantial research (Müller and Scheichl (2014); Maynard et al (2020); Kühnlein et al (2019); Piotrowski et al (2016); Dedner et al (2016)). There have been some advances to apply machine-learning methods within the context of linear solvers.…”

Krylov solvers in a vertical‐slice version of the semi‐implicit semi‐Lagrangian AROME model

“…Krylov methods are also used for example in Smolarkiewicz and Margolin (1994), where the Generalized Conjugate Residual (GCR) method is used successfully in a density-stratified potential flow past a steep three-dimensional isolated hill on a plane, or in Thomas, et al (1997) where a preconditioned Generalized Minimal Residual (GMRES) method solver is successfully used in a global SI-SL model, very close to the AROME model. More recently, Kühnlein, et al (2019) and Maynard, et al (2020) use preconditioned Krylov solvers in the context of, respectively, a conserving finite-volume model and a finite-element discretization. All these Krylov iterative methods seem particularly well suited since they mainly require local operations (as in HEVI schemes), the numbers of which depend on the accuracy required and the speed of convergence to achieve it.…”

“…(2019) and Maynard, et al . (2020) use preconditioned Krylov solvers in the context of, respectively, a conserving finite‐volume model and a finite‐element discretization. All these Krylov iterative methods seem particularly well suited since they mainly require local operations (as in HEVI schemes), the numbers of which depend on the accuracy required and the speed of convergence to achieve it.…”