  The paper is devoted to the Lie group properties of the one-dimensional
Green-Naghdi equations describing the behavior of fluid flow over uneven bottom
topography. The bottom topography is incorporated into the Green-Naghdi
equations in two ways: in the classical Green-Naghdi form and in the
approximated form of the same order. The study is performed in Lagrangian
coordinates which allows one to find Lagrangians for the analyzed equations.
Complete group classification of both cases of the Green-Naghdi equations with
respect to the bottom topography is presented. Applying Noether's theorem, the
obtained Lagrangians and the group classification, conservation laws of the
one-dimensional Green-Naghdi equations with uneven bottom topography are
obtained. Difference schemes which preserve the symmetries of the original
equations and the conservation laws are constructed. Analysis of the developed
schemes is given. The schemes are tested numerically on the example of an exact
traveling-wave solution.
