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

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