中文摘要：

在辐射流体力学的数值模拟中,扩散算子的高效高精度离散是一个十分重要的问题.本文研究各向异性扩散方程在任意多边形网格上的数值求解问题,我们利用调和平均点和线性精确方法,构造了一个单元中心型有限体积格式.该格式只含有单元中心未知量,满足局部守恒条件,有紧凑的计算模板,在结构四边形网格上退化为一个九点格式.由于调和平均点插值算法是一个具有两点模板的二阶保正算法,因此,采用单元边上的调和平均点为插值节点,使得离散格式十分简洁,容易实施.此外,我们在格式构造中仅采用了二、三维网格的共有拓扑关系,使格式容易向三维问题推广,大部分程序代码可实现二、三维公用.我们采用典型的大变形扭曲网格及典型的扩散算例（包括连续和间断的扩散张量）对所提出的新格式进行了测试,数值算例表明,新格式在许多扭曲的多边形网格上具有二阶精度.

英文摘要：

An accurate and effective discretization of diffusion operators is very important in some practical applications such as radiation hydrodynamics. In this paper, we discuss the numerical solution of anisotropic diffusion problems on arbitrary polygonal meshes. A cell- centered finite volume scheme is constructed based on the harmonic averaging point through a certain linearity-preserving approach. The new scheme has only cell-centered unknowns, is locally conservative and has a compact stencil, which reduces to a nine-point scheme on structured quadrilateral meshes. Since the interpolation algorithm based on the harmonic averaging point is a two-stencil and positivity-preserving one, the construction of the scheme is largely simplified. Moreover, since we only use the common topology of 2D and 3D meshes, the extension of the new scheme to the 3D case is very easy and most of codes can be shared. In numerical experiments, we employ some typical distorted meshes and diffusion problems with both continuous and discontinuous coefficients to test our scheme. Numerical results show that the new scheme has a second-order accuracy on many distorted polygonal meshes.