Numerical models for integrating partial-differential equations of weather and climate (W&C) often involve elliptic boundary value problems (BVPs). Solution of such BVPs amounts to inverting a sparse N x N matrix, with N being a total number of nodes discretizing the W&C model domain. With numerical weather prediction aiming at horizontal resolutions O(1)~km globally, N can be on the order of 10⁹. The best suited methods for handling such large linear problems are the matrix-free Krylov-subspace algorithms, with the conjugate gradient scheme being a prominent example.  Krylov schemes may be viewed as pseudo-time integrations of the damped oscillations equations, with a variable time step and damping scales determined variationally by minimizing the solution departures from the solution attainable in the stationary limit of the successive iterations. Notwithstanding their virtues of simplicity and robustness, Krylov schemes lose the efficacy for stiff problems characterized by the multiplicity of scales. To aid their performance, while retaining the virtues, the vanilla schemes incorporate operator preconditioning that amounts to solving efficiently a part of the governing BVP problem and reduce its stiffness.  

The present lecture discusses a suite of preconditioners steaming from the Richardson's (1910) seminal 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 (as opposed to variational) iteration solvers, including the many classical algorithms. The high-performance all-scale EULAG model [J. Comput. Phys. 263 (2014) 185-205], 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.

Given the availability of the article underlying this lecture [J. Comput. Phys. 463 (2022) 111296], I will only highlight key theoretical aspects of the BVPs in W&C models and focus on revealing the relative efficiency and accuracy gains attainable with the developed preconditioners in simulations of atmospheric flows across scales from planetary to cloud and laboratory.