浓度对流扩散方程高精度差分格式的构造及其在环境中的应用

发布时间：2018-03-28 13:13

本文选题：浓度对流扩散方程　切入点：高精度　出处：《大连海事大学》2017年硕士论文

【摘要】：在现实的科学技术中,多数流体类问题都可以用浓度对流扩散方程来描述,如:污染物在水和大气中的分布以及行为归宿,都可以用浓度对流扩散方程描述。因此,构造稳定、高效、精度高的浓度对流扩散方程的求解算法,有着极为重要的理论和实际应用意义。目前较常用的数值解法有:有限体积法、有限元法、有限分析法以及有限差分法。在工程领域和科学研究中最为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网格点少,精度高的优点,成为众多学者研究的热点问题。本文通过两种方法构造具有高精度的差分格式,采用Von Neumann分析法及数值计算对其稳定性进行分析,并将其应用于环境问题当中。首先,在本文的第二章中通过待定系数法,针对浓度扩散方程和浓度对流扩散方程,构造了三层高精度差分格式,其精度可达到O(△t4,△x8)。通过引用相关数值算例进行数值计算,对数值解和精确解进行对比,发现二者数值基本一致,且二者间的误差可以达到理论误差,即本文所构造的三层差分格式有效且可以达到理论精度。其次,在本文的第三章中,将浓度关于时间的一阶偏导数在时间层n+1/2处进行离散。将空间n+1/2处的二阶偏导数,用第n+1和n时间层的空间二阶偏导数的平均值表示。为使空间上达到更高的精度,将浓度在空间上进行泰勒级数展开,进而构造两层高精度差分格式。当泰勒级数展开到第N项时,其精度可达到O(△t2,△xN)。通过数值算例验证本文所构造的两层差分格式有效,且可以达到理论精度。最后,在本文的第四章。以围油栏结合收油装置处理溢油问题为例,采用本文所构造的高精度差分格式,对油浓度的变化进行数值模拟,进而根据收油装置单位时间内的额定收油量,选择合适的收油速度。
[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.
【学位授予单位】：大连海事大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8;X55

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