A Semi-Lagrangian and Semi-Implicit Two Time-Level Integration Scheme

Abstract

A semi-Lagrangian and semi-implicit two time-level integration scheme has been constructed for integrating the primitive meteorological equations. It can be thought of as working in the following way. During the first half-time step the Coriolis terms are integrated implicitly, while the pressure gradient terms are integrated explicitly. During the second half-time step the Coriolis terms are integrated explicitly, while the pressure gradient terms are integrated implicitly. The advection terms are integrated by means of a multiply-upstream semi-Lagrangian scheme and the nonlinear terms are integrated explicitly, once per time step. The scheme is shown to be unconditionally stable when the equations of motion are linearized about an isothermal basic state. It is also very efficient because the implicit integrations can either be solved directly (in the case of the Coriolis terms) or give rise to a Helmholtz equation for which efficient fast solvers exist (in the case of the pressure gradient terms). A real-time run of this scheme in parallel with the Irish Meteorological Service's operational forecast model showed no deterioration in forecast quality while cutting the required CPU by more than 60%.