本文選題：濃度對流擴(kuò)散方程　切入點(diǎn)：高精度　出處：《大連海事大學(xué)》2017年碩士論文

【摘要】：在現(xiàn)實(shí)的科學(xué)技術(shù)中,多數(shù)流體類問題都可以用濃度對流擴(kuò)散方程來描述,如:污染物在水和大氣中的分布以及行為歸宿,都可以用濃度對流擴(kuò)散方程描述。因此,構(gòu)造穩(wěn)定、高效、精度高的濃度對流擴(kuò)散方程的求解算法,有著極為重要的理論和實(shí)際應(yīng)用意義。目前較常用的數(shù)值解法有:有限體積法、有限元法、有限分析法以及有限差分法。在工程領(lǐng)域和科學(xué)研究中最為常用的是有限差分法,而具有高精度的有限差分法以其具有涉及網(wǎng)格點(diǎn)少,精度高的優(yōu)點(diǎn),成為眾多學(xué)者研究的熱點(diǎn)問題。本文通過兩種方法構(gòu)造具有高精度的差分格式,采用Von Neumann分析法及數(shù)值計(jì)算對其穩(wěn)定性進(jìn)行分析,并將其應(yīng)用于環(huán)境問題當(dāng)中。首先,在本文的第二章中通過待定系數(shù)法,針對濃度擴(kuò)散方程和濃度對流擴(kuò)散方程,構(gòu)造了三層高精度差分格式,其精度可達(dá)到O(△t4,△x8)。通過引用相關(guān)數(shù)值算例進(jìn)行數(shù)值計(jì)算,對數(shù)值解和精確解進(jìn)行對比,發(fā)現(xiàn)二者數(shù)值基本一致,且二者間的誤差可以達(dá)到理論誤差,即本文所構(gòu)造的三層差分格式有效且可以達(dá)到理論精度。其次,在本文的第三章中,將濃度關(guān)于時(shí)間的一階偏導(dǎo)數(shù)在時(shí)間層n+1/2處進(jìn)行離散。將空間n+1/2處的二階偏導(dǎo)數(shù),用第n+1和n時(shí)間層的空間二階偏導(dǎo)數(shù)的平均值表示。為使空間上達(dá)到更高的精度,將濃度在空間上進(jìn)行泰勒級數(shù)展開,進(jìn)而構(gòu)造兩層高精度差分格式。當(dāng)泰勒級數(shù)展開到第N項(xiàng)時(shí),其精度可達(dá)到O(△t2,△xN)。通過數(shù)值算例驗(yàn)證本文所構(gòu)造的兩層差分格式有效,且可以達(dá)到理論精度。最后,在本文的第四章。以圍油欄結(jié)合收油裝置處理溢油問題為例,采用本文所構(gòu)造的高精度差分格式,對油濃度的變化進(jìn)行數(shù)值模擬,進(jìn)而根據(jù)收油裝置單位時(shí)間內(nèi)的額定收油量,選擇合適的收油速度。
[Abstract]:In practical science and technology, most fluid problems can be described by concentration convection-diffusion equations, such as the distribution and behavior of pollutants in water and atmosphere, which can be described by concentration convection-diffusion equations. It is very important to solve the convection-diffusion equation of concentration with high efficiency and precision, which is of great theoretical and practical significance. At present, the commonly used numerical methods are: finite volume method, finite element method, finite element method, finite volume method, finite element method, finite volume method and finite element method. The finite difference method is the most commonly used method in the field of engineering and scientific research. The finite difference method with high accuracy has the advantages of less mesh points and higher precision. In this paper, the difference scheme with high precision is constructed by two methods. The stability of the scheme is analyzed by Von Neumann analysis and numerical calculation, and it is applied to environmental problems. In the second chapter of this paper, based on the undetermined coefficient method, a three-layer high-precision difference scheme is constructed for the concentration diffusion equation and the concentration convection diffusion equation. The accuracy of the scheme can reach O (t _ 4, x _ 8). By comparing the numerical solution with the exact solution, it is found that the numerical value is basically the same, and the error between them can reach the theoretical error, that is, the three-layer difference scheme constructed in this paper is effective and can achieve the theoretical accuracy. Secondly, in the third chapter of this paper, The first-order partial derivative of concentration with respect to time is discretized at time layer n 1 / 2. The second order partial derivative of space n 1 / 2 is represented by the mean value of space second order partial derivative of n 1 and n time layer. The Taylor series expansion of concentration is carried out in space, and a two-layer high-precision difference scheme is constructed. When the Taylor series is expanded to the N term, its accuracy can reach O (t _ 2, x _ N _ n). The numerical examples show that the two-layer difference scheme constructed in this paper is effective. Finally, in the fourth chapter of this paper, the oil concentration change is numerically simulated by using the high-precision difference scheme constructed in this paper, taking the oil spill problem treated by the oil containment column and the oil recovery device as an example. Furthermore, according to the rated oil recovery rate per unit time of the oil recovery unit, the appropriate oil recovery rate is selected.
【學(xué)位授予單位】：大連海事大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.8;X55

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