Development, implementation, and verification of hp discontinuous Galerkin models for shallow water hydrodynamics and transport.

摘要

Shallow water hydrodynamic and transport equations are used to describe many free surface flow and transport processes in the deep ocean, coastal ocean, estuaries, rivers, open channels, and coastal floodplain. In many practical applications, these equations must be solved on domains with geometrically complex vertical and horizontal boundaries introduced by both the bathymetry/topography and the coastline. Finite element methods are a natural choice for such problems given the ease with which unstructured grids can be implemented; however, there are well known problems associated with solving these equations using the standard Galerkin method.; The discontinuous Galerkin (DG) finite element method offers a solution strategy to solve the numerical difficulties associated with these equations. The main advantages of DG methods for shallow water flow and transport are: their ability to capture smooth physically damped solutions to wave propagation problems; their ability to handle advection dominated flows and sharp gradients including problems with hydraulic jumps or bores (discontinuities); their inherent elemental mass and momentum conservation properties, which make them ideal for coupling flow and transport models; and the ease with which both h (grid) and p (polynomial order) refinement, and also adaptivity can be implemented.; In this dissertation, robust, accurate, computationally efficient, and flexible one- and two-dimensional hp DG finite element models for shallow water hydrodynamics, passive transport, and sediment transport are developed, implemented, and verified. The performance of the models is demonstrated and rigorously assessed by examining a number of test cases including linear and highly nonlinear problems, both uncoupled and coupled transport problems, idealized coastal modeling applications, problems with discontinuities or shocks, and full-scale applications. Through systematic h and p convergence studies the dual paths to convergence of the method are demonstrated for both linear and highly nonlinear problems where it is observed that standard h-refinement for a fixed p leads to optimal convergence rates of p + 1, while exponential convergence is observed for p-refinement on a fixed mesh. Additionally, the efficiency and effectiveness of h- and p-refinement, a variety of time stepping schemes, and a simple p-adaptive algorithm are investigated with the goal of obtaining highly accurate solutions in the most efficient manner possible.