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 analogs 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.

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Actually, on a fixed segment axb, the value may differ up to a constant, which depends on H.

70.

See the details on this well-known method and its stability conditions, for example, in Ref. 68. It is easy to verify that the stability conditions of the algorithm are satisfied for system (6.10).

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