Consistent, conservative, and efficient advection updates for iterative-implicit atmospheric solvers

Abstract

Atmospheric model dynamical cores that iterate towards a Crank-Nicolson-like implicit time stepping scheme are attractive for operational prediction because their excellent stability properties permit the use of long time steps. However, the long-timestep advection schemes used in such models are relatively expensive, and that expense is compounded by the need to compute the advection terms multiple times in the iterative solver. Moreover, unless care is taken in the design of the solver, desirable properties of an advection scheme such as conservation, consistency, and boundedness might only be achieved in the unaffordable limit of solver convergence. Here a modification to such iterative solvers is proposed, similar to the previously published SLIC scheme, in which full advection calculations are made only once per time step, with cheap advection updates made at each solver iteration. This modification significantly reduces the cost of such iterative solvers. It is shown here that the cheap advection updates and the solver back-substitution calculations can be formulated in such a way that the advection remains conservative, consistent, and bounded no matter how many solver iterations are taken, and not only at solver convergence. The proposed approach is demonstrated in shallow-water model simulations.