  The one-dimensional shallow water equations in Eulerian coordinates are
considered. Relations between symmetries and conservation laws for the
potential form of the equations, and symmetries and conservation laws in
Eulerian coordinates are shown. An invariant difference scheme for equations in
Eulerian coordinates with arbitrary bottom topography is constructed. It
possesses all the finite-difference analogues of the conservation laws. Some
bottom topographies require moving meshes in Eulerian coordinates, which are
stationary meshes in mass Lagrangian coordinates. The developed invariant
conservative difference schemes are verified numerically using examples of flow
with various bottom topographies.
