本文選題：橢圓型方程 + 混合型��； 參考：《寧夏大學》2017年碩士論文

【摘要】：本文主要針對一般橢圓型方程,構(gòu)造了兩種混合型高精度緊致差分格式.首先,基于泰勒級數(shù)展開,推導了一維橢圓型方程的四階混合型緊致差分格式,緊接著在四階差分格式的基礎(chǔ)上,依然采用泰勒級數(shù)展開的余項修正方法,得到了求解一維橢圓型方程的六階混合型緊致差分格式.并分析了兩種差分格式的截斷誤差,結(jié)果表明兩種差分格式的理論精度分別為四階和六階.最后通過一些具有精確解的算例進行了數(shù)值驗證,并與其它格式進行了數(shù)值結(jié)果對比,突顯出了本文格式的優(yōu)越性.對于二維問題,在一維格式的基礎(chǔ)上分別推導了求解二維橢圓型方程的四階和六階混合型緊致差分格式,并且分析了兩種差分格式的截斷誤差,然后通過一些具有精確解的算例進行了數(shù)值驗證,并與其它格式進行了數(shù)值結(jié)果對比.針對三維問題,本文推導了四階混合型緊致差分格式,并分析了格式的截斷誤差,最后,通過一些具有精確解的算例進行了數(shù)值驗證,并與其它格式進行了數(shù)值結(jié)果對比.本文所研究的方程模型具有一般性,尤其對于定常對流擴散方程而言,本文格式能夠很好地數(shù)值模擬大雷諾數(shù)問題,這也是本文格式相較于其他格式的一大優(yōu)點.
[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.
【學位授予單位】：寧夏大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.82

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本文編號：1968149