椭圆型方程的混合型高精度紧致差分格式
发布时间:2018-06-02 08:45
本文选题:椭圆型方程 + 混合型 ; 参考:《宁夏大学》2017年硕士论文
【摘要】:本文主要针对一般椭圆型方程,构造了两种混合型高精度紧致差分格式.首先,基于泰勒级数展开,推导了一维椭圆型方程的四阶混合型紧致差分格式,紧接着在四阶差分格式的基础上,依然采用泰勒级数展开的余项修正方法,得到了求解一维椭圆型方程的六阶混合型紧致差分格式.并分析了两种差分格式的截断误差,结果表明两种差分格式的理论精度分别为四阶和六阶.最后通过一些具有精确解的算例进行了数值验证,并与其它格式进行了数值结果对比,突显出了本文格式的优越性.对于二维问题,在一维格式的基础上分别推导了求解二维椭圆型方程的四阶和六阶混合型紧致差分格式,并且分析了两种差分格式的截断误差,然后通过一些具有精确解的算例进行了数值验证,并与其它格式进行了数值结果对比.针对三维问题,本文推导了四阶混合型紧致差分格式,并分析了格式的截断误差,最后,通过一些具有精确解的算例进行了数值验证,并与其它格式进行了数值结果对比.本文所研究的方程模型具有一般性,尤其对于定常对流扩散方程而言,本文格式能够很好地数值模拟大雷诺数问题,这也是本文格式相较于其他格式的一大优点.
[Abstract]:In this paper, two mixed compact difference schemes with high precision are constructed for general elliptic equations. First of all, based on Taylor series expansion, the fourth order mixed compact difference scheme of one-dimensional elliptic equation is derived, and then the remainder correction method of Taylor series expansion is adopted on the basis of fourth-order difference scheme. A six order mixed compact difference scheme for solving one dimensional elliptic equations is obtained. The truncation errors of the two schemes are analyzed. The results show that the theoretical accuracy of the two schemes is four and six respectively. Finally, some numerical examples with exact solutions are given, and the numerical results are compared with those of other schemes, which show the superiority of this scheme. For the two-dimensional problem, based on the one-dimensional scheme, the four-order and sixth-order compact difference schemes for solving two-dimensional elliptic equations are derived, and the truncation errors of the two schemes are analyzed. Then the numerical results are verified by some examples with exact solutions, and the numerical results are compared with other schemes. In this paper, the fourth order mixed compact difference scheme is derived, and the truncation error of the scheme is analyzed. Finally, some numerical examples with exact solutions are given and compared with other schemes. The equation model studied in this paper is general, especially for the steady convection-diffusion equation, the scheme in this paper can simulate the problem of large Reynolds number numerically, which is one of the advantages of the scheme compared with other schemes.
【学位授予单位】:宁夏大学
【学位级别】:硕士
【学位授予年份】:2017
【分类号】:O241.82
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