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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 ›› 2020, Vol. 37 ›› Issue (6): 719-729.doi: 10.3969/j.issn.1005-3085.2020.06.006

• • 上一篇    下一篇

非等距网格上一维对流扩散方程的保正格式

兰   斌1,2,   王   涛1   

  1. 1- 北方民族大学数学与信息科学学院，银川  750021
    2- 宁夏智能信息与大数据处理重点实验室，银川 750021
  • 收稿日期:2019-02-25 接受日期:2019-12-10 出版日期:2020-12-15 发布日期:2021-02-15
  • 基金资助:
    国家自然科学基金 (11601013)；宁夏自然科学基金 (2019AAC03129)；北方民族大学自然科学基金 (2019XYZS03).

Positivity-preserving Scheme of 1D Convection-diffusion Equation on Nonuniform Meshes

LAN Bin1,2,   WANG Tao1   

  1. 1- School of Mathematics and Information Sciences, North Minzu University, Yinchuan 750021
    2- The Key Laboratory of Intelligent Information and Big Data Processing of Ningxia Province, Yinchuan 750021
  • Received:2019-02-25 Accepted:2019-12-10 Online:2020-12-15 Published:2021-02-15
  • Supported by:
    The National Natural Science Foundation of China (11601013); the Natural Science Foundation of Ningxia (2019AAC03129); the Natural Science Foundation of North Minzu University (2019XYZS03).

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摘要： 对流扩散方程广泛存在于很多领域，为适应一些实际问题模型的求解，对离散格式，不仅要求满足一些基本性质，如稳定性和解的存在唯一性等，还要求离散格式的保正性．采用有限体积格式求解对流扩散方程的工作较少，但在保正性方面所做的工作不多．本文构造了任意非等距网格上一维对流扩散方程的非线性保正有限体积格式．其中，扩散通量的离散，在等距网格上，当扩散系数为标量时可退化为标准的二阶中心差分格式．而对流通量的离散，为避免数值振荡而使其保持迎风特性，提出一种新的方法使格式精度提高到二阶．该方法在上游单元中心处作泰勒级数展开，通过相关辅助未知量来完成梯度的重构，并对出负情形作正性校正，使得格式满足保正性要求．新格式只含有区间单元中心未知量，并满足区间端点处通量的局部守恒性．数值结果表明，本文所提格式是有效的，对于处理扩散占优、对流占优问题，扩散系数连续和间断情形均具有良好的适应性，并且保持二阶精度．另外，新格式适用于扩散系数间断问题的求解．

关键词: 对流扩散方程, 非等距网格, 有限体积格式, 保正性

Abstract: The convection diffusion equation exists widely in many fields. In order to solve some practical problems, the discretization scheme should not only satisfy some basic properties, such as convergence, stability and the existence and uniqueness of solutions, but also keep the positivity of the discretization scheme. A lot of researches has been done to solve the convection diffusion equation by using the finite volume scheme, but little work has been done in the aspect of keeping the positivity. In this paper, a nolinear positivity-preserving finite volume scheme for the one-dimensional convection diffusion equation on arbitrary nonuniform grids is constructed. The scheme is unformed in a matrix form. Then, it is proved that the scheme satisfies the requirement of positivity-preserving by using the properties of the coefficient matrix. The scheme only contains the unknown quantity of the center of the interval element and satisfies the local conservation of flux at the end of the interval. Finally, the numerical results show that the proposed scheme is effective and owns the second order accuracy. In addition, the scheme is applicable to the solution of problems with discontinuous diffusion coefficients.

Key words: convection diffusion equation, nonuniform meshes, finite volume scheme, positivity-preserving

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