Finite Volume Method On Unstructured Meshes And The Study Of Moving Mesh Based On Deformation Method

It is difficult to obtain analytic solution of conservation laws because of its quasi-linearity generally, which promote the development of numerical methods. Since 1950's, the research of numerical methods for hyperbolic conservation laws, specially TVD and ENO schemes, are very successful. Because of the solution of hyperbolic conservation laws changing rapidly only in fairly localized regions, grid adaptation in hyperbolic conservation laws has attracted much attention. Considering the reasons aforementioned, the following work has been done in this thesis:1. In chapter 2, we construct a class of non-oscillatory finite volume method for hyperbolic conservation laws on unstructured meshes. We get the linear interpolation by sing the least square method and ensure that the solution will not produce the new extremum by using the maximum principle. Finally we construct the non-oscillatory numerical scheme. The method avoids choosing template, needs less computation and has high resolution.2. Chapter 3 mainly presents moving mesh based on deformation method and its applications in hyperbolic conservation laws. Recently, grid adaptation is extensively studied and exploited, which is used to advance resolution of numerical solution. And the technology has become an important numerical tool in a variety of physical and engineering fields. This paper develops the moving mesh method based on deformation presented by Liao to triangular meshes, and gives the updated equation from old mesh to new mesh in accurate form, and then we apply the method to 2-D hyperbolic conservation laws. The experimental results denote that this method is efficient.3. In chapter 4, we modify the mesh generated in chapter 3. Concerning about error brought by practical computation, we define quality parameter to evaluate the quality of moving mesh method based on deformation, and modify the newly obtained grid. The modified method is applied into practical examples. The numerical results denote that modification method is feasible and efficient.