The Discontinuous Galerkin Method For Solving Shallow Water Equations

Solving the one-dimensional and two-dimensional shallow water equations by using the discontinuous Galerkin method is the main subject in this thesis. We give detailed information on methods or procedures that are involved in using the discontinuous Galerkin method.Base on the quadtree grid generation method, we developed a hybrid type grid generation method , which is a mix of the structured grid generation method and the unstructured grid generation method.A new searching algorithm and a new method for managing grid memory is implemented in our grid generation method. The main advantages of our method is that : it fits well on rectangular area ; it allows to have multiple level of hanging nodes and doesn't need to do grid regularization; it is high efficiency and robust. This method was applied to the problem of oblique hydraulic jump, two-dimensional rectangular and circular dam-break problems; An adaptive procedure is applied and good result is achieved.The numerical fluxes and limiters that are suited for the discontinuous Galerkin method were also discussed. We apply different type of numerical fluxes alternatively after a fixed period of time. This kind of method worked well in our numerical cases. We also find a numerical flux which is not of Godunov type and which matches well with the viscous flow. We analyze the characters and functions of several kinds of limiters , and give a well tuned computing parameter for a modified slope limiter. We compare several numerical fluxes and limiters on the one dimensional dam-brek problem , and the finite difference method is also compared to the discontinuous Galerkin method.We give a discretization method for the viscous terms of the two-dimensional shallow water equations , and the obtained discretized equations are used to solve the backward-facing step flow problem.From the numerical cases in the thesis, we can see that the discontinuous Galerkin method could generate more sharp result;the combination with adaptive grid generation method showed that our method could accurately capture local small scale features. They also showed that the discontinuous Galerkin method is suitable for general shallow water free-surface problems.