Galerkin Methods For Two Dimensional Shallow Water Equations

The shallow water equations have a wide range of applications in the earth'satmosphere, ocean, environmental and hydraulic engineering, and clean energyexploitation. Related issues include tsunamis and storm surges prediction, sedi-ment and contaminant transport, tidal energy capturing in estuaries and coastalregions, etc. Based on unstructured meshes, the finite element approaches canhandle complex geometry with ease, therefore they have been developed rapidlyin the numerical study of shallow water waves over the past three decades.There are some recent advances, such as characteristic-Galerkin method, non-linear Galerkin method, local discontinuous Galerkin method and space-timediscontinuous Galerkin method.In this thesis, we mainly study two types of Galerkin finite element methodfor the two-dimensional depth-averaged shallow water equations, which includethe MMOCAA-Galerkin method and the coupled discontinuous and continuousGalerkin method.The MMOC (modified method of characteristics) is a time-stepping proce-dure based on the characteristics, the core of which is to combine the time deriva-tive and the advection term as a directional derivative along the characteristics.The MMOC allows for larger time steps than those of standard time-steppingmethods without the loss of accuracy, and eliminates the excessive non-physicaloscillation and numerical dispersion as well. Unfortunately, the MMOC can notpreserve mass conservation. To eliminate the mass balance error in the MMOC,a variant of the MMOC, called the MMOCAA, was presented by Douglas etal. ([25, 26]). By imposing a higher-order perturbation on the foot of character-istics, the MMOCAA does preserve the desired conservation property and alsothe conceptual and computational advantages of the MMOC. In chapter 3, Wepresent the MMOCAA-Galerkin formulations for the two-dimensional shallow water equations. It is shown that, the scheme yields suboptimal-order errorestimates for elevation and velocity in∞(0,T);L2(?) and an optimal-orderestimate of O(h + ?t) for velocity in 2 (0,T);H1(?) . These estimates pre-serve the same accuracy as the MMOC-Galerkin formulations (Dawson et al.,[19]). Furthermore, according to the algorithm analysis, the scheme improvesglobal mass conservation at a minor additional computational cost.Discontinuous Galerkin methods possess a number of favorable properties,such as the ability to incorporate stable and higher-order approximations, the?exibility with respect to hp-adaptivity, and the local conservation property.However, compared with continuous Galerkin methods, discontinuous Galerkinmethods have much larger amount of calculation. Recently, Dawson et al. ([20–22,38]) have investigated a coupled discontinuous and continuous Galerkin methodfor the shallow water equations. Its basic idea (see [21]) is to use discontinuousGalerkin method where the solution might have sharp fronts or local conservationis important, and to use continuous Galerkin method where the solution is rela-tively smooth, so as to balance between e?ciency and performance. In chapter 4,we employ the coupled method in [20] to a more complex shallow water system,mainly supplementing the analysis of the nonlinear convection term and externalforces. The coupling strategy adopted is to use discontinuous Galerkin methodfor the primitive continuity equation and to use continuous Galerkin method forthe nonconservative momentum equations.