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

A limited-area model of linearized shallow water equations (SWE) on an f -plane for a rectangular domain is considered. The rectangular domain is extended to include the so-called perfectly matched layer (PML) as an absorbing boundary condition. Following the proponent of the original method, the equations are obtained in this layer by splitting the shallow water equations in the coordinate dir...

Abstract We derive the Whitham equations from water waves in shallow regime using two different methods, thus obtaining a direct and rigorous link between these models. The first one is based on construction of approximate Riemann invariants for Whitham–Boussinesq system adapted to unidirectional waves. second generalisation Birkhoff’s normal form algorithm almost smooth Hamiltonians bidirectio...

In this paper we consider the solution of the enhanced Boussinesq equations of Madsen and Sørensen (Coast.Eng. 18, 1992) by means of residual based discretizations. In particular, we investigate the applicability of upwind and stabilized variants of the Residual Distribution and Galerkin finite element schemes for the simulation of wave propagation and transformation over complex bathymetries. ...