  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.
