Preconditioning elliptic operators in high-performance all-scale atmospheric models on unstructured meshes

Effective simulation of all-scale atmospheric flows – e.g., cloud-resolving global weather – involves semi-implicit integration of the non-hydrostatic compressible Euler equations under gravity on a rotating sphere. Such integrations depend on complex non-symmetric elliptic solvers. The condition number of the underlying sparse linear operator is $O\left({10}^{10}\right)$, which necessitates bespoke operator preconditioning. This paper highlights the development and implementation on unstructured meshes of specialised preconditioners for the non-symmetric Krylov-subspace solver. These developments are set in the context of a massively-parallel high-performance computing environment, aimed at architectures evolving towards exascale.

The baroclinic instability benchmark bearing representative features relevant to numerical weather prediction (NWP) has been selected to study the performance of the preconditioning options. The reported results illustrate the improved performance with the new preconditioning options. In particular, the Jacobi based option, for the computational meshes tested in this study, provides an excellent time to solution improvement.