The second sense of the term is used about algorithms, that converge even for very large integration steps (relative to other algorithms employed for the same problem). Hence, if only the steady-state is of interest an, in this sense, convergent algorithm is preferred. The pressure-velocity couplings, such as PISO and SIMPLE are normally viewed as convergent, while the ADI methods such as Peaceman-Rachford, Briley-McDonald, and Beam-Warming are not convergent, according to this definition. Convergence in this sense may not be proved for the Navier-Stokes equations. Experience, however, shows that each algorithm has a certain range of parameters, for example the CFL number, within which they perform near-optimally.
Dan N. Sorensen