Lie group symmetry analysis and invariant difference schemes of the two-dimensional shallow water equations in Lagrangian coordinates

The two-dimensional shallow water equations in Lagrangian coordinates are considered. Lie group classification for the class of elliptic paraboloid bottom topography is performed. transformations mapping with a plane or rotation symmetric into gas dynamics polytropic exponent γ=2 represented. foliation discussed. New invariant conservative finite-difference schemes and their one-dimensional reductions constructed. derived either by extending known direct algebraic construction based on some assumptions form energy conservation law. Among proposed there possessing laws mass energy.