An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows

In atmospheric flow situations typical of a small-scale atmospheric thermal, a separation of time scales exists between the fast sound wave time scale and the advective time scale. Atmospheric models have been designed to take advantage of this disparity of time scales with numerical approaches such as the semi-implicit or split-explicit approach being used to efficiently step over the fast sound waves. Some of these numerical approaches are first order in time. To improve accuracy over these methods, a fully implicit and nonlinearly consistent (INC) flow solver has been developed for the Navier–Stokes equation set. In our INC method, the equation set is solved by use of the Jacobian-free Newton–Krylov (JFNK) method. An efficient preconditioner has been developed which uses the semi-implicit method to solve the governing equations. Being that this preconditioner was designed to attack the fastest waves in the system and not other features in the implicit system such as advection or turbulent diffusion, the preconditioning technique is labeled as a physics-based preconditioner. A variety of linear solvers including SSOR, Krylov methods and/or multigrid approaches are used to approximately invert the pressure matrix in the semi-implicit algorithm. A suite of simulations will be conducted utilizing different linear solvers for the simple problem of the bouyant rise of a warm bubble. The problem will also document the ability of the INC approach to achieve second order in time accuracy.