任意多边形网格上扩散问题的一个新型有限体积格式  

A New Finite Volume Scheme for Diffusion Problems on Arbitrary Polygonal Meshes

作　　者：单丽 张振 SHAN LI;ZHANG ZHEN(College of Science,Shantou Uriversitxy,Shantou 515063,China;School of Mathematics and Statistics,lluazhong University of Science and Technology,Whan.430074,China)

机构地区：[1]汕头大学理学院,汕头515063 [2]华中科技大学数学与统计学院,武汉430074

出　　处：《应用数学学报》2020年第6期1042-1053,共12页Acta Mathematicae Applicatae Sinica

基　　金：国家科学自然基金(11401284);中国国家留学基金(201808210153)资助项目。

摘　　要：在多介质辐射流体力学的数值模拟中,研究扩散算子的高精度离散格式是一个十分重要的问题.本文在任意多边形网格上针对扩散问题建立了一个高精度的单元中心型的有限体积格式.我们选择调和平均点和网格边两端的节点作为辅助插值点,这些辅助插值点的场变量可表示为网格中心点场边量的线性组合,通过解欠定线性方程组来确定权重系数,最终得到只含单元中心未知量的离散格式.该格式满足局部守恒条件,在结构四边形上退化为一个九点格式.此外,我们的格式容易推广至三维情形.在随机四边形网格,三角形网格和Shestakov网格上,我们针对不同类型场变量函数的进行数值实验,均可得到二阶的收敛速度.An accurate and effective discretization of diffusion operators is very important in some practical applications such as radiation hydrodynamics.In this paper,a highprecision cell-centered finite volume method is constructed for the diffusion problem on arbitrary polygonal meshes.We choose the harmonic averaging point and two end points of mesh edge as the auxiliary interpolation points,which can be expressed as a linear combination of the central unknowns,hence the cell-centered scheme follows by solving the corresponding underdetermined linear equations.Our scheme maintains local conservation property,it is easy to degenerate into a nine-point scheme on a structural quadrilateral mesh.Moreover,our scheme can be easily extended to three dimensions cases.On the randomly perturbed mesh,acute triangular mesh and Shestakov mesh,we perform the numerical tests for different types of field variable functions,second-order convergence rates are achieved.

关 键 词：任意多边形网格 有限体积方法 扩散问题 线性精确 调和平均点 欠定线性方程组 

分 类 号：O242.2[理学—计算数学]