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

Abstract

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