The Finite Volume Schemes For Three Dimensional Diffusion Equations

The finite volume method is a sort of discrete method which is used in the numerical simulation of all kinds of conservation problems (elliptic, parabolic and hyperbolic problems). It has a wide range of applications in the practical work such as the fluid mechanics, heat conduction, petroleum engineering and so on. The main idea is as follows:we integral firstly the original equations at the discrete mesh which is called the control volume. Then we get the integral form of the flux at the edge of control volume using the flux theorem. At last, the discrete format will be got by the discretization of the boundary flux. The finite volume method is applicable to structure of non-structure meshes, and it has the superiority because of the special quality of the local conservation of the numerical flux.The rest of this paper is organized as follows.In chapter1, we introduce the related physics background and the prelim-inaries. And we compare the finite volume method with the finite difference method and finite element method.In chapter2, we elaborate the development and the deduction of four kinds of cell-centered finite volume schemes, including nine points schemes, SOM schemes, multi-point flux approximating and monotone schemes.Chapter3, which is the vital content of this paper, we put forward a kind of finite volume scheme on the arbitrary three dimensional arbitrary hexahe-dron meshes, and prove its monotonicity.