Stability of Semi-Implicit and Iterative Centered-Implicit Time Discretizations for Various Equation Systems Used in NWP

The stability of classical semi-implicit scheme, and some more advanced schemes recently proposed for Numerical Weather Prediction (NWP) is examined. In all these schemes, the solution of the centred-implicit-linear equation is approached by an iterative fixed-point algorithm, by a simple, constant in time, linear operator. A general for assessing analytically the stability of these schemes on problems for a vertically unbounded atmosphere is presented. The method is valid for all the equation systems usually employed in NWP., as in earlier studies, the method can be applied only in simplified contexts, thus overestimating the actual stability that would in more realistic meteorological contexts. The analysis is performed in spatially-continuous framework, hence allowing to eliminate the-discretisation or the boundary conditions as possible causes of the instabilities linked to the time-scheme itself. The general method then shown concretely to apply to various time-discretisation schemes and-systems (namely shallow-water, and fully compressible Euler). Analytical results found in the literature are recovered from the method, and some original results are presented.