Conservative invariant finite‐difference schemes for the modified shallow water equations in Lagrangian coordinates

Abstract

AbstractThe one‐dimensional modified shallow water equations in Lagrangian coordinates are considered. Relations between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables and Eulerian coordinates are discussed. For equations in Lagrangian coordinates, invariant finite‐difference schemes are constructed. These schemes admit all conservation laws, which are related to Lie symmetries admitted by the differential model via Noether's theorem: they possess the difference analogs of the conservation laws of mass, momentum, energy, the law of center‐of‐mass motion for horizontal, inclined, and parabolic bottom topographies. The invariant conservative difference schemes are tested numerically in comparison with an ad hoc approximation‐invariant scheme.