A Least Square Finite Volume Method For 2D Shallow Water Equations On Unstructured Meshes

The Shallow Water Equations arises in many applications such as oceanic, environment and bionomics. It is well known that the solutions of the Shallow Water Equations can be discontinuous even when the initial condition is smooth. This thesis is concerned with the least square finite volume method for Shallow Water Equations on unstructured meshes.The thesis consists of five chapters.In chapter 1, we briefly review the physical background, mathematical model and numerical methods for the Shallow Water Equations.In chapter 2, we present the numerical schemes for resolving Shallow Water Equations on structured and unstructured meshes, such as Godunov and composite schemes.In chapter 3, we present frame of the finite volume methods for 2-D hyperbolic conservation laws. Also we introduce control volume, TVD Runge-Kutta time discretization, maximum principle and limiters.In chapter 4, we present the non-oscillatory finite volume method for the Shallow Water Equations on unstructured meshes. We get the linear interpolation using the least square method and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the non-oscillatory numerical scheme. The method avoids choosing template, needs less computation and has high resolution of the discontinue.In chapter 5, we present the third-order finite volume method for the Shallow Water Equations on unstructured meshes. We get the secondary interpolation using the least square method by enlarging the template once and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the third-order high resolution numerical scheme. It can capture discontinuous well.At last, we summarize the advantages and disadvantages in chapter4, 5, and show our task in the future.