The Radial Basis Function Method For Hyperbolic Conservation Laws

Hyperbolic conservation laws is extensively used in hydromechanics, aerodynamics, Aircraft Industries, biology and many other engineering areas. Generally, there is no analysis solution for it, so a large body of deeper study is inspired and some kinds of numerical method are developed, such as finite difference method (FDM), finite element method (FEM), finite volume method (FVM), and meshless method. The primary advantages of finite volume method are numerical robustness, applicability on general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. So finite volume method is a class of discretization schemes which have been proved to be highly successful in approximating the solution of a wide variety of conservation law systems, some examples include computational fluid mechanics, meteorology, computational electromagnetics, semi-conductor device simulation, and computational biology. Meshless method is a numerical method in which the physical variables are approximated at a finite number of nodes instead of the meshes as in finite volume method, thus it completely or partly eliminates the meshes.The following work has been done in this dissertation:1. An ENO finite volume method which is constructed on the basis of radial basis functions on unstructured triangular meshes is introduced. In order to obtain the higher accuracy on spatial discretization, an interpolation radial basis function is constructed on every triangular mesh. Two point Gauss quadrature formula is also used on every edge of every triangular mesh and the third order TVD Runge-Kutta method is used for time discretization. Numerical experiments show that the method compute fast and improve resolving power of the discontinuous domain.2. We present a moving mesh method based on spring principium and apply it in solving Euler equations successfully. We give the updated map from old meshes to new meshes by using a weighted reconstruction method which satisfies conservation laws.3. The basic theory of meshless method and its applications for computational fluid dynamics are investigated in this dissertation. The spatial discretization is estimated using radial basis functions. The paper adopts explicit four-steps Runge-Kutta scheme for time discretization to solve the discrete form for Euler equations. Numerical experiments show that the method is effective.