非定常扩散方程的单元中心型有限体积格式  

Cell-centered Finite Volume Scheme for Evolutionary Diffusion Equation

作　　者：单丽 张海成 金珠 SHAN Li;ZHANG HAICHENG;JIN ZHU(Department of Mathematics,Shantou University,Shantou 515063,China;School of Mathematical Sciences,East China Normal University,Shanghai 200241,China;College of Sciences,Liaoning Technical University,Fuain 123000,China)

机构地区：[1]汕头大学数学系,汕头515063 [2]华东师范大学数学科学学院,上海200241 [3]辽宁工程技术大学理学院,阜新123000

出　　处：《应用数学学报》2023年第1期73-87,共15页Acta Mathematicae Applicatae Sinica

基　　金：汕头大学科研启动基金(批准号:NTF21006);辽宁省教育厅科学研究基金(批准号:LJ2020JCL009)资助项目。

摘　　要：本文基于调和平均点建立了一种新的单元中心型有限体积格式,用以求解非定常扩散方程.在网格边上离散法向流时,选择该网格边两端点和该边上的一个调和平均点作为辅助插值点,并将这些辅助插值点上的未知量用网格单元中心点的未知量进行替换,最终得到一个只含网格单元中心未知量的有限体积格式.该格式满足线性精确性质和局部守恒性,且适用于任意多边形网格.在六种不同的多边形网格上进行四个数值实验,分别考虑扩散系数是连续的和间断的以及非线性的情况,数值结果表明:本文所构造的格式在六种网格上的L2误差均可达到二阶收敛精度,对于不同类型的扩散系数,该格式保持良好的鲁棒性,并且从编程实现的角度来说,该格式更易于向三维情况推广.By introducing the harmonic averaging point,we propose a new cell-centered finite volume scheme for evolutionary diffusion equation in this paper.On a mesh edge,we choose its two vertices and a harmonic averaging point as the auxiliary interpolation points.To make the finite volume scheme a cell-centered one,we replace the unknowns on these auxiliary points by the unknowns on the central point of the corresponding grid elements.Our scheme is linearity-preserving and local conservative,it can be applied on an arbitrary polygonal mesh.Considering the cases that the diffusion coefficient is continuous,discontinuous and nonlinear,respectively,four numerical experiments are implemented on six different polygonal meshes.The numerical results show that our scheme is second-order convergent in L2 norm,it maintains good robustness for different types of diffusion coefficients,besides,it is easy to extend to 3D cases in programming.

关 键 词：有限体积方法 非定常扩散方程 任意多边形网格 线性精确 调和平均点 

分 类 号：O241.82[理学—计算数学]