Publications
5 results found
Betteridge JDD, Cotter CJJ, Gibson THH, et al., 2023, Hybridised multigrid preconditioners for a compatible finite-element dynamical core, Quarterly Journal of the Royal Meteorological Society, Vol: 149, Pages: 2454-2476, ISSN: 0035-9009
Compatible finite-element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging, since the velocity mass matrix is nondiagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate and weather prediction model LFRic.
Ham DA, Kelly PHJ, Mitchell L, et al., 2023, Firedrake user manual, Firedrake User Manual
Betteridge JD, Farrell PE, Ham DA, 2021, Code generation for productive, portable, and scalable finite element simulation in Firedrake, Computing in Science and Engineering, Vol: 23, Pages: 8-17, ISSN: 1521-9615
Creating scalable, high performance PDE-based simulations requires a suitablecombination of discretizations, differential operators, preconditioners andsolvers. The required combination changes with the application and with theavailable hardware, yet software development time is a severely limitedresource for most scientists and engineers. Here we demonstrate that generatingsimulation code from a high-level Python interface provides an effectivemechanism for creating high performance simulations from very few lines of usercode. We demonstrate that moving from one supercomputer to another can requiresignificant algorithmic changes to achieve scalable performance, but that thecode generation approach enables these algorithmic changes to be achieved withminimal development effort.
Betteridge J, 2021, GenX Workshop: Planning for exascale continuum mechanics software
Betteridge J, Gibson TH, Graham I, et al., 2020, 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.
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