Hybridised implicit solvers for the Gung Ho dynamical core

Abstract: Gung Ho is the name of the Met Office project to build a new dynamical core (fluid dynamics component) for their weather/climate prediction system. Gung Ho is built around compatible finite element methods as the apparently unique solution to the question of how to find a consistent gridpoint (i.e. non-spectral) discretisation that supports various essential wave propagation properties at the discrete level on grids with near-equal edge lengths "pseudo-uniform" on the sphere. One downside of this approach versus their current finite difference approach is the non-diagonal mass matrix for the velocity component, which means that the usual strategy of eliminating velocity to get an elliptic problem for pressure results in a non-sparse matrix. The solution to this, known for decades, is to "hybridise" the mixed system by breaking continuity constraints to get a discontinuous velocity space, and to introduce Lagrange multipliers as trace variables supported on cell facets to enforce continuity of the solution. The system can then be eliminated down to a sparse reduced system for the trace variables only. The question then arises of how to efficiently iteratively solve this system when the domain is very thin (like the Earth's atmosphere). This question can be answered by combining various results from (a) the analysis of hybridised mixed finite element methods and (b) the analysis of additive Schwarz methods. I will briefly introduce these, describe a solver algorithm and sketch a proof that it gives iteration counts that are independent of depth in the thin layer limit, before illustrating with some numerical results produced using Firedrake and PETSc.

computational engineering, finance, and sciencemathematical softwarenumerical analysiscomputational physics

Audience: researchers in the topic

Australian Seminar on Computational Mathematics

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