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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 ›› 2023, Vol. 40 ›› Issue (3): 439-455.doi: 10.3969/j.issn.1005-3085.2023.03.008

• • 上一篇    下一篇

辐射扩散方程基于虚拟元方法的一个保正守恒格式

盛美花1,   杨  堤1,   高志明2   

  1. 1. 中国工程物理研究院研究生院，北京 100088
    2. 北京应用物理与计算数学研究所，北京 100088
  • 收稿日期:2020-12-30 接受日期:2021-03-29 出版日期:2023-06-15 发布日期:2023-08-15
  • 基金资助:
    国家自然科学基金 (11771052).

A Positivity-preserving and Conservative Scheme Based on the Virtual Element Method for Radiation Diffusion Equations

SHENG Meihua1,  YANG Di1,   GAO Zhiming2   

  1. 1. Graduate School of China Academy of Engineering Physics, Beijing 100088
    2. Institute of Applied Physics and Computational Mathematics, Beijing 100088
  • Received:2020-12-30 Accepted:2021-03-29 Online:2023-06-15 Published:2023-08-15
  • Supported by:
    The National Natural Science Foundation of China (11771052).

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摘要：

作为近年来广受关注的一种数值方法，虚拟元方法具有很多优势。但在求解实际问题导出的一些辐射扩散方程时，该方法可能无法保证数值解的非负性及一般多边形网格上的局部守恒性。针对辐射扩散方程，利用非线性两点流逼近方法作为后处理措施，提出了一种基于虚拟元方法的保正守恒格式。该格式通过最低阶虚拟元方法得到数值解的单元顶点值，再利用非线性两点流逼近方法得到数值解的非负单元中心值，同时使格式满足局部守恒性。任意多边形网格上的数值结果表明，该格式具有保正性和解的近似二阶收敛速度，对于处理含强间断或非线性扩散系数的辐射扩散问题均有较强的适应性。

关键词: 辐射扩散方程, 虚拟元方法, 保正性, 局部守恒性, 非线性两点流逼近

Abstract:

As a widely adopted numerical method in recent years, the virtual element method has many advantages. However, when solving some radiation diffusion equations derived from practical problems, the method may not guarantee the non-negativity of the numerical solution or maintain the local conservation property on general polygonal meshes. This paper uses the nonlinear two-point flux approximation as a post-processing procedure, and proposes a positivity-preserving and conservative scheme based on the virtual element method for radiation diffusion equations. The scheme obtains the cell-vertex values of the numerical solution by the lowest-order virtual element method. Then the positive cell-centered values are obtained by the nonlinear two-point flux approximation, where the local conservation property is maintained as well. The numerical results on arbitrary polygonal meshes demonstrate the second-order convergence rate for the solution scheme, and its high adaptability to deal with radiation diffusion problems with strong discontinuous or nonlinear diffusion coefficients.

Key words: radiation diffusion equation, virtual element method, positivity-preserving property, local conservation property, nonlinear two-point flux approximation

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