Unstructured Grid Large Particle Finite Volume Method

Since 1950' s, the research of numerical method for hyperbolic conservation laws is one of key research directions in computational mathematics. The finite difference method, specially TVD and ENO schemes, for hyperbolic conservation laws are very successful. But the finite difference methods require that the computational domain is regular, and usually people construct the finite difference scheme for one dimension and then extend it to two or three dimensions in a dimension by dimension fashion. Due to the engineering problem and computational domain become more complicated and more complicated, the finite volume method on unstructured meshes for hyperbolic conservation laws is playing an important role in computational fluid dynamics, this method has no restrctions for computational domain. In this paper,we consider hyperbolic conservation laws, and the following work has been done to the finite volume method on unstructured meshes :In preface, we introduce the backupground of the oringe of the finite volume method, and point out the limitation of the finite difference methods and the finite element methods, and import the finite volume method. Then introduce the oringe and develepment of the finite volume methods, and give some results of the new methods.In the chapter 2, we give the base frame of the finite volume methods, that is meshes splitting,discrete the space and discrete the time.In the chapter 3, we give a series of new type finite volume methods on unstructured meshes. And these methods represent the direction of the research of the finite volume methods.In the chapter 4 ,we give a simplified WENO(weighted essentially nonoscillatory)scheme on unstructured triangular meshes for two dimensions scalar hyperbolic conservtion law. In order to approximate the flux, a weighted quadratic reconstruction is given. The resulting, simplified WENO scheme is based on cell average and a TVD Runge-kutta time discretization. Preliminary encouraging numerical experiment is given.In the chapter 5,we present a three order accuracy Large Particle finite volume method on unstructured triangular meshes for 2D invisid Euler equations. The computational scheme consists of two steps for each time march step. An "ENO" type quadratic interpolation is ultilized on every mesh and the "Riemann" solution is found at the first step. Finally, the numerical results for plate shock reflection and blunt cylinder flow are satisfactory.