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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 ›› 2016, Vol. 33 ›› Issue (6): 578-586.doi: 10.3969/j.issn.1005-3085.2016.06.002

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基于Leonard规正变量的修正高分辨率组合格式

牛云霞1,2, 李春光3, 景何仿3   

  1. 1. 北方民族大学数学与信息科学学院，银川 750021
    2. 新疆大学科学技术学院(阿克苏校区)，阿克苏 843000
    3. 北方民族大学数值计算与工程应用研究所，银川 750021
  • 收稿日期:2015-04-07 接受日期:2015-12-01 出版日期:2016-12-15 发布日期:2017-02-15
  • 通讯作者: 李春光 E-mail: cglizd@hotmail.com
  • 基金资助:
    国家自然科学基金 (11361002)；宁夏自治区水利厅水资源项目 ([2015]50-18)；北方民族大学研究生创新项目 (YCX1555)；北方民族大学数学与信息科学学院研究生创新项目.

The Modified High-resolution Composite Scheme Based on the Leonard Normalized Variable

NIU Yun-xia1,2, LI Chun-guang3, JING He-fang3   

  1. 1. School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021
    2. Xinjiang University Institute of Science and Technology (Akesu Campus), Akesu 843000
    3. Research Institute of Numerical Computation and Engineering Applications, Beifang University of Nationalities, Yinchuan 750021
  • Received:2015-04-07 Accepted:2015-12-01 Online:2016-12-15 Published:2017-02-15
  • Contact: C. Li. E-mail address: cglizd@hotmail.com
  • Supported by:
    The National Natural Science Foundation of China (11361002); the Ningxia Water Resources Project ([2015]50-18); the Graduate Student Innovation Projects of Beifang University of Nation-alities (YCX1555); the Graduate Student Innovation Projects of School of Mathematics and Information Science of Beifang University of Nationalities.

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摘要： 通过对规正变量进行重构，本文提出了求解对流扩散方程的修正高分辨率组合格式，它能够求解边界层和大梯度等问题．首先，根据规正变量的定义得出了组合格式的通用表达式，然后对时间项采用二阶中心差分格式，得到了对流扩散方程的离散表达式，对离散化得到的代数方程组采用TDMA算法求解，并推导出了组合格式计算过程迭代收敛时所满足的充分条件．数值实验表明：新格式具有分辨率高，数值耗散较低，总偏差量较小，能很好模拟场变量的大梯度变化，计算结果优于传统格式．

关键词: 对流扩散方程, 高分辨率组合格式, Leonard规正变量, TDMA算法

Abstract: Based on the normalized variable, a modified high-resolution composite scheme is proposed to solve the convection-diffusion equation, which can be used to describe boundary layer problems or locally large gradient problems. Firstly, the generic form for the composite scheme is derived according to the definition of the normalized variable. Then, the second order central difference scheme is used to discretize the temporal derivative term. The discrete algebraic equations are solved with the TDMA algorithm. Thus, the sufficient condition for the convergence of the composite scheme is obtained. The numerical tests are presented to verify the effectiveness of the scheme. The numerical results show that the new format has higher resolution, lower numerical dissipation and smaller total amount of deviation when compared with traditional formats. The new scheme is good at simulating the large gradient field variables.

Key words: convection-diffusion equation, high-resolution combination scheme, Leonard normalized variable, TDMA algorithm

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