The local discontinuous Galerkin method for three-dimensional shallow water flow

We formulate and present a stability analysis for the local discontinuous Galerkin method applied to a model of three-dimensional shallow water flow. This model is described by the Navier–Stokes equations, assuming hydrostatic pressure. The resulting system of equations is given by momentum equations for the x and y components of velocity, the continuity equation, which is used to solve for the vertical component of velocity, and an equation describing the motion of the free surface. Our analysis includes the full nonlinearities in the model with no simplifying assumptions and accounts for the movement of the mesh due to the free surface. To our knowledge this is the first analysis of any numerical scheme for this complex system of equations. Numerical results are presented which demonstrate the accuracy of the method for a problem with a known analytical solution.