  Invariant discretization schemes are derived for the one- and two-dimensional
shallow-water equations with periodic boundary conditions. While originally
designed for constructing invariant finite difference schemes, we extend the
usage of difference invariants to allow constructing of invariant finite volume
methods as well. It is found that the classical invariant schemes converge to
the Lagrangian formulation of the shallow-water equations. These schemes
require to redistribute the grid points according to the physical fluid
velocity, i.e., the mesh cannot remain fixed in the course of the numerical
integration. Invariant Eulerian discretization schemes are proposed for the
shallow-water equations in computational coordinates. Instead of using the
fluid velocity as the grid velocity, an invariant moving mesh generator is
invoked in order to determine the location of the grid points at the subsequent
time level. The numerical conservation of energy, mass and momentum is
evaluated for both the invariant and non-invariant schemes.
