  We consider the outward-propagating nonlinear concentric water waves within
the scope of the 2D Boussinesq system. The problem is axisymmetric, and we
derive the slow radius versions of the cylindrical Korteweg - de Vries (cKdV)
and extended cKdV (ecKdV) models. Numerical runs are initially performed using
the full axisymmetric Boussinesq system. At some distance away from the origin,
we use the numerical solution of the Boussinesq system as the "initial
condition" for the derived cKdV and ecKdV models. We then compare the evolution
of the waves as described by both reduced models and the direct numerical
simulations of the axisymmetric Boussinesq system. The main conclusion of the
paper is that the extended cKdV model provides a much more accurate description
of the waves and extends the range of validity of the weakly-nonlinear
modelling to the waves of moderate amplitude.
