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The paper is devoted to the Lie group properties of the one-dimensionalGreen-Naghdi equations describing the behavior of fluid flow over uneven bottomtopography. The bottom topography is incorporated into the Green-Naghdiequations in two ways: in the classical Green-Naghdi form and in theapproximated form of the same order. The study is performed in Lagrangiancoordinates which allows one to find Lagrangians for the analyzed equations.Complete group classification of both cases of the Green-Naghdi equations withrespect to the bottom topography is presented. Applying Noether's theorem, theobtained Lagrangians and the group classification, conservation laws of theone-dimensional Green-Naghdi equations with uneven bottom topography areobtained. Difference schemes which preserve the symmetries of the originalequations and the conservation laws are constructed. Analysis of the developedschemes is given. The schemes are tested numerically on the example of an exacttraveling-wave solution.

Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations