求解对流扩散方程的一致四阶紧致格式

doi:10.12286/jssx.2016.4.391

计算数学 ›› 2016, Vol. 38 ›› Issue (4): 391-404.DOI: 10.12286/jssx.2016.4.391

求解对流扩散方程的一致四阶紧致格式

王涛^1,2, 刘铁钢¹

1. 北京航空航天大学数学与系统科学学院, 北京 100191;
2. 北方民族大学预科教育学院, 银川 750021

收稿日期:2015-10-23 出版日期:2016-12-15 发布日期:2016-10-13
基金资助:
国家自然科学基金资助项目（No.91130019.No.10931004 and No.11601013）.

PDF ( 1018 ) 引用被引次数 ( 7 | 1 )

王涛, 刘铁钢. 求解对流扩散方程的一致四阶紧致格式[J]. 计算数学, 2016, 38(4): 391-404.

Wang Tao, Liu Tiegang. A CONSISTENT FOURTH-ORDER COMPACT SCHEME FOR SOLVING CONVECTION-DIFFUSION EQUATION[J]. Mathematica Numerica Sinica, 2016, 38(4): 391-404.

A CONSISTENT FOURTH-ORDER COMPACT SCHEME FOR SOLVING CONVECTION-DIFFUSION EQUATION

Wang Tao^1,2, Liu Tiegang¹

1. School of Mathematics and Symtems Science, Beihang University, Beijing 100191, China;
2. College of Preparatory Education, Northern university for Nationalities, Yinchuan 750021, China

Received:2015-10-23 Online:2016-12-15 Published:2016-10-13

目前，许多高精度差分格式，由于未成功地构造与其精度匹配的稳定的边界格式，不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式，它的边界点的计算格式和内点的计算格式的截断误差主项保持一致，给出了具体内点和边界格式；并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度；在对流占优情况下，本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小，取得了不错的效果，理论结果得到了数值验证；驱动方腔数值结果显示，本文对N-S方程的离散格式具有很好的可靠性，适合对复杂流体流动的数值模拟和研究.

关键词： 一致四阶格式, 渐近稳定性, 对流占优, 边界层

At present, many high accuracy difference scheme, due to not successfully constructed and its matching accuracy and stable boundary format, have to adopt the low accuracy of boundary format. This paper proves the existence of convection diffusion equation to be consistent fourth-order compact schemes and the boundary point calculation format is consistent with the truncation error of additional qualifications and gives the concrete inner point and boundary scheme; the asymptotic stability of discrete scheme is analyzed. Numerical example shows that the scheme is fourth order accuracy; under the condition of convection-dominant, numerical example shows that the numerical oscillation that is achieved by the boundary scheme of this paper is smaller than four order accuracy of explicit difference scheme, good results have been achieved and theoretical results obtained numerical validation.The simulation results of the regularized driven cavity are good agreement with the existing solutions in the literature and shows the discrete scheme of N-S equation has a good reliability, suitable for the research of complex fluid flow.

Key words: consistent fourth-order boundary scheme, the asymptotic stability, convection-dominated, boundary layer

MR(2010)主题分类:

65D17

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摘要

求解对流扩散方程的一致四阶紧致格式

A CONSISTENT FOURTH-ORDER COMPACT SCHEME FOR SOLVING CONVECTION-DIFFUSION EQUATION

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