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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (5): 797-812.doi: 10.3969/j.issn.1005-3085.2022.05.009

• • 上一篇    下一篇

任意多边形上非定常扩散方程的单元中心型有限体积格式

单   丽1,   金   珠2,   张海成3   

  1. 1. 汕头大学理学院，汕头  515063;   2. 辽宁工程技术大学理学院，阜新  123000;   3. 华东师范大学数学科学学院，上海  200241
  • 出版日期:2022-10-15 发布日期:2022-12-15
  • 基金资助:
    辽宁省教育厅科学研究基金(LJ2020JCL009)；汕头大学科研启动基金(NTF21006).

Cell-centered Finite Volume Scheme for Evolutionary Diffusion Equations on Arbitrary Polygonal Meshes

SHAN Li1,   JIN Zhu2,    ZHANG Haicheng3   

  1. 1. College of Science, Shantou University, Shantou 515063;
    2. School of Science, Liaoning Technical University, Fuxin 123000;
    3. School of Mathematical Sciences, East China Normal University, Shanghai 200241
  • Online:2022-10-15 Published:2022-12-15
  • Supported by:
    The Scientific Research Foundation of Education Department of Liaoning Province (LJ2020JCL009); the Scientific Research Foundation for Talents of Shantou University (NTF21006).

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摘要：

针对二维非定常扩散方程，构造适用于任意多边形网格的单元中心型有限体积格式。采用向后欧拉格式进行时间离散，空间上在离散扩散算子时，利用网格顶点作为辅助插值点，通过求解一个欠定方程组将辅助插值点信息替换成网格单元中心点信息，最终得到只含单元中心未知量的离散格式。该格式既满足局部守恒条件，又满足线性精确准则。在几类多边形网格上进行数值实验，分别考虑扩散系数是连续和间断的情况，发现新格式均可达到二阶收敛。其数值表现显著优于算数平均加权和逆距离加权的九点格式，与双线性插值的加权方式结果相近，并且克服了双线性插值加权方式不适用于三角形网格的弊端。数值算例表明新格式求解非线性扩散方程仍然可以达到二阶收敛。

关键词: 有限体积方法, 非定常扩散方程, 多边形网格, 线性精确, 单元中心型

Abstract:

A cell-centered finite volume scheme for the 2D evolutionary diffusion equation on arbitrary polygonal meshes is constructed. We apply the backward Euler scheme to discrete the time derivative term, and employ the vertex unknowns as auxiliary ones to discrete the diffusion operator, by solving an underdetermined linear system of equations, vertex unknowns can be expressed by a linear combination of the central unknowns, which finally results in a cell-centered scheme. The proposed scheme maintains the local conservation and the linearity preserving properties. Considering the continuous and discontinuous diffusion coefficients respectively, several numerical experiments on different kinds of polygonal meshes show that second-order convergence rate can be obtained. Its numerical performance is significantly better than the nine point scheme with arithmetic average weighting and inverse distance weighting, and is similar to the weighting method of bilinear interpolation, it overcomes the disadvantage that bilinear interpolation is not suitable for triangular meshes. Besides, the numerical results also implies that proposed scheme can still achieve second-order convergence for solving nonlinear diffusion equations.

Key words: finite volume method, evolutionary diffusion equation, polygonal mesh, linearity-preserving, cell-centered

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