A suite of Richardson preconditioners for semi-implicit all-scale atmospheric models

Abstract

The paper documents a suite of preconditioners for Krylov-subspace solvers of elliptic boundary-value problems (BVPs) that underlie semi-implicit integrations of the nonhydrostatic equations governing the dynamics of all-scale atmospheric flows. Effective preconditioning of the linear operators inherent in the semi-implicit models lies at the heart of the state-of-the-art multiscale-flow simulation. This is especially evident in simulations of global weather and climate—posed on a thin spherical shell—where some form of direct tridiagonal inversion of the operator in the vertical is crucial to relax the often enormous stiffness of the problem. The documented preconditioners stem from the Richardson's (1910) idea of augmenting an elliptic BVP with a transient diffusion equation. Exploiting this idea for mixed explicit-implicit pseudo-time-stepping schemes leads to a broad suite of stationary-iteration solvers, including the many classical algorithms. Here, the high-performance all-scale EULAG model (Smolarkiewicz et al. (2014) [58]), with a flexible three-dimensional decomposition of MPI tasks, is furnished with the preconditioners akin to the classical alternating-direction-implicit (ADI) algorithms, generalized to optional permutations of parallel tridiagonal inversions. The utility of various options is found to be problem dependent, in terms of computational accuracy as well as efficiency. The main thrust of the work is on the long-range forecasts using large anisotropic grids. The relative efficiency and/or accuracy gains attainable with the developed preconditioners are illustrated for idealized scenarios representative of atmospheric flows from planetary to a single-cloud and laboratory scales. The key insight that best encapsulates the significance and novelty of the present work is that there is no single “super-preconditioner” that will perform best in all cases, yet the suite as a whole offers substantial gains in the model performance.