A time splitting scheme for the nonhydrostatic atmospheric model

A semi-implicit finite difference scheme with time splitting corresponding to the multi-scale nature of atmospheric dynamics is developed for the nonhydrostatic atmospheric model. The fast acoustic and gravity waves are approximated implicitly, while relatively slow advective terms and Rossby modes are treated explicitly. The used level of implicitness requires a solution of the three-dimensional linear elliptic equation at each time step, which is accomplished efficiently by vertical decoupling and subsequent application of fast and accurate multigrid solvers for the most significant vertical modes, and fast and simpler solvers for the secondary ones. Analysis of the linear stability of the scheme reveals that the Courant–Friedrichs–Lewy condition on the time step associated with gravity and acoustic waves can be circumvented and the maximum stable time step can be brought into closer agreement with the time step limitations arising from accuracy considerations. The performed numerical experiments show the computational efficiency of the designed scheme and accuracy of the predicted atmospheric fields.