Runge - Kutta Control Volume For Solving Hyperbolic Conservation Laws With Discontinuous Finite Element Method (rkcvdfem)

In the numerical simulation of fluid dynamics, study of numerical methods for hyperbolic conservation laws is very important. It has already become the third means for investigating problems of fluid dynamics besides theory and experiment. Study of numerical methods greatly promotes the development of fluid dynamics. In the numerical simulation of practical fluid dynamics, in order to obtain exact and sharp physical image, study of the schemes with high order and high resolution is a key and also a significative topic.In this thesis, we study numerical methods for hyperbolic conservation laws. Enlightened by Runge-Kutta discontinuous Galerkin finite element method (RKDGM), we use control volume finite element method (CVFEM) to solve hyperbolic conservation laws in the first time and devise a new high order and high resolution scheme, which we call it Runge-Kutta control volume discontinuous finite element method (RKCVDFEM). The dissertation includes the scheme construction, theoretical analysis and numerical experiments. New method combines the good qualities of RKDGM and CVFEM, which can easily handle the discontinuous solution and boundary conditions, keep local conservation, compute simplicity and so on. The spatial discretization is based on CVDFEM and the temporal discretization is based on TVB Runge-Kutta technique. We construct three different schemes to solve hyperbolic conservation laws and discuss these schemes' properties; we obtain the convergent results. By using of RKCVDFEM, we test the typical examples of hyperbolic conservation laws. The results show high order and high resolution of the scheme and verify our theoretical analysis.