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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 ›› 2017, Vol. 34 ›› Issue (3): 283-296.doi: 10.3969/j.issn.1005-3085.2017.03.005

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求解变系数对流扩散反应方程的指数型高精度紧致差分方法

田   芳,    葛永斌   

  1. 宁夏大学数学统计学院，银川  750021
  • 收稿日期:2015-06-08 接受日期:2016-09-07 出版日期:2017-06-15 发布日期:2017-08-15
  • 基金资助:
    宁夏大学自然科学基金(ZR15014).

An Exponential High Accuracy Compact Finite Difference Method for the Convection-diffusion-reaction Equation with Variable Coefficients

TIAN Fang,   GE Yong-bin   

  1. School of Mathematics and Statistics, Ningxia University, Yinchuan 750021
  • Received:2015-06-08 Accepted:2016-09-07 Online:2017-06-15 Published:2017-08-15
  • Supported by:
    The Natural Science Foundation of Ningxia University (ZR15014).

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501

被引次数

1

摘要：

本文给出了一种数值求解变系数对流扩散反应方程的指数型高精度紧致差分方法．我们首先将模型方程变形，借助常系数对流扩散方程的指数型高精度紧致差分格式，采用残量修正法得到变系数对流扩散反应方程的指数型高精度紧致差分格式；并从理论上分析了当Pelect数很大时，本文格式达到四阶计算精度时网格步长的限制条件；离散得到的代数方程组可采用追赶法直接求解．数值实验结果与理论分析完全吻合，表明了本文格式对于边界层问题或大梯度变化的物理量求解问题具有的高精度和鲁棒性的优点．

关键词: 对流扩散反应方程, 指数型有限差分格式, 高精度紧致差分格式, 对流占优, 边界层

Abstract:

An exponential high accuracy compact finite difference method is proposed to solve the one-dimension (1D) convection-diffusion-reaction equation with variable coefficients. Fir-stly, the equation is rewritten in the form of convection diffusion equation. Then the exponential high order compact finite difference scheme for the convection diffusion equation with constant coefficients and the remainder term modification approach are utilized to obtain an exponential high accuracy compact finite difference scheme for the 1D convection-diffusion-reaction equation with variable coefficients. Secondly, the necessary condition on grid step length is analyzed theoretically if the scheme in this paper has a fourth-order accuracy when the Peclet number is very high. Lastly, the Thomas approach is applied to deal with the algebraic equations. Numerical examples, mostly with the boundary layer where sharp gradients may appear due to high Peclet number, are presented to demonstrate the accuracy and robustness of the proposed scheme.

Key words: convection-diffusion-reaction equation, exponential finite difference scheme, high accuracy compact difference scheme, convection dominant, boundary layer

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