Shallow water modeling using discontinuous and coupled finite element methods

Abstract

This chapter discusses two approaches for shallow water flow modeling based on discontinuous and continuous approximating spaces. In the first approach, it discretises the primitive continuity equation using a discontinuous Galerkin (DG) method, coupled to a continuous finite element approximation of the momentum equations. This approach is useful when local conservation is important and it uses discontinuous approximations for the hyperbolic continuity equation while allowing the momentum equation to be approximated through more traditional and continuous functions. In the second approach, it discretises both equations using DG methods. In both approaches, a DG method is used to approximate the continuity equation. This DG approach has several appealing features, particularly the ability to incorporate upwinding and stability postprocessing into the solution to model highly advective flows, the ability to use different polynomial orders of approximation in different parts of the domain (and for different variables, if so desired), and the ability to easily use nonconforming meshes. The shallow water equations (SWE) model flow in the domains whose characteristic wavelength in the horizontal is much larger than the water depth. The SWEs consist of a first-order hyperbolic continuity equation for the water elevation, coupled to momentum equations for the horizontal depth-averaged velocities.