本文選題：非定常對流擴散方程　切入點：有理型高精度緊致格式　出處：《寧夏大學(xué)》2015年碩士論文

【摘要】：本文主要建立了求解對流擴散方程的有理型高精度緊致(RHOC)差分方法.首先在空間上,基于函數(shù)的泰勒級數(shù)展開和空間四階緊致差分公式,推導(dǎo)了一維定常對流擴散方程的RHOC差分格式.然后,在時間上利用Crank-Nicolson格式進行離散,得到了求解一維非定常對流擴散方程的RHOC差分格式,該差分格式時間上具有二階精度,空間上具有四階精度.通過vonNeumann分析方法證明了RHOC格式是無條件穩(wěn)定的.通過與其它幾種已有格式的數(shù)值算例結(jié)果比較,驗證了RHOC格式的精確性和穩(wěn)定性.接著,基于一維問題的研究,分別推導(dǎo)出了二維和三維非定常對流擴散方程的交替方向隱式的有理型高精度緊致(RHOC ADI)差分格式,該差分格式時間上具有二階精度,空間上具有四階精度,并且是無條件穩(wěn)定的.數(shù)值實驗結(jié)果表明,本文針對非定常對流擴散方程所建立的RHOC ADI差分方法,不僅能夠適用于非定常對流擴散問題,而且能夠很好地求解非定常純對流問題或純擴散問題,并且其計算效果均優(yōu)于其它的差分格式.該方法很好地結(jié)合了高精度緊致差分格式和ADI方法的優(yōu)勢,為求解非定常對流擴散方程提供了一類精確、穩(wěn)定、高效的數(shù)值方法.最后,推導(dǎo)了一維定常對流擴散反應(yīng)方程的RHOC差分格式,并利用Richardson外推法和算子插值技術(shù)將格式的精度提高到六階.并通過數(shù)值實驗驗證了格式的精確性和可靠性.
[Abstract]:In this paper, the rational high precision compact RHOC difference method for solving convection-diffusion equations is established. Firstly, the Taylor series expansion based on function and the fourth order compact difference formula in space are established. The RHOC difference scheme for one dimensional steady convection-diffusion equation is derived, and then the RHOC difference scheme for solving one dimensional unsteady convection-diffusion equation is obtained by using Crank-Nicolson scheme in time. The difference scheme has second order accuracy in time. It is proved that the RHOC scheme is unconditionally stable by the vonNeumann analysis method. The accuracy and stability of the RHOC scheme are verified by comparison with the numerical examples of other schemes. Based on the study of one-dimensional problems, the alternating direction implicit rational compact RHOC ADI difference schemes for two-dimensional and three-dimensional unsteady convection-diffusion equations are derived respectively. The scheme has second order accuracy in time. The numerical results show that the RHOC ADI difference method for unsteady convection-diffusion equations can not only be applied to unsteady convection-diffusion problems. Moreover, the unsteady pure convection problem or pure diffusion problem can be solved well, and its calculation results are superior to those of other difference schemes. This method combines the advantages of the high precision compact difference scheme and the ADI method. A kind of accurate, stable and efficient numerical method is provided for solving unsteady convection-diffusion equation. Finally, the RHOC difference scheme of one-dimensional steady convection-diffusion reaction equation is derived. The Richardson extrapolation method and operator interpolation technique are used to improve the accuracy of the scheme to the sixth order, and the accuracy and reliability of the scheme are verified by numerical experiments.
【學(xué)位授予單位】：寧夏大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O241.82

【參考文獻】

相關(guān)期刊論文 前1條

1 趙秉新;;一維非定常對流擴散方程的高階組合緊致迎風(fēng)格式[J];數(shù)值計算與計算機應(yīng)用;2012年02期

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