浓度对流扩散方程高精度并行格式的构造及其应用

发布时间：2018-11-26 16:27

【摘要】：对流扩散方程在环境领域中有重要的作用,比如污染物质在河流、大气等中的分布情况,我们可以用对流扩散方程来描述。所以,对对流扩散方程求解方法的研究具有极为重要的意义。目前,在对流扩散方程的求解过程中,有限差分方法是众多数值求解方法中重要的方法之一。但是,该方法中的显格式稳定性受到限制；隐格式需解大型的线性方程组,难以并行。近些年来,随着高性能计算机的快速发展,并行算法发展迅速,国内外研究学者取得了众多研究成果。例如,Evans和张宝琳等人针对扩散方程和对流扩散方程提出了一系列的并行算法。但是众多并行算法中的有些精度并不高,所以,构造高精度并行算法,以达到提高计算速度和精度的目的是很有必要的。本文在隐格式的基础上,主要是对一维浓度对流扩散方程和二维浓度对流扩散方程构造了新的并行算法,并且将二维浓度扩散方程高精度并行算法应用到环境的实际问题中。在本文的第二章中,针对一维、二维浓度扩散方程,一维、二维浓度对流扩散方程,构造了一类新的二阶精度隐式并行格式,并分别对各个算例进行了计算,对计算结果进行分析,验证了格式的有效性。在本文的第三章中,针对一维浓度扩散方程、一维浓度对流扩散方程及二维浓度扩散方程,分别构造了一类新的高阶精度并行格式,并分别对各个算例进行了计算,对数值解和精确解进行比较,发现两者的曲线完全吻合,计算精度明显高于二阶精度隐式并行格式的计算精度,证明了格式的有效性。在本文的第四章中,以二维浓度扩散方程的高精度并行格式为理论基础,研究水产养殖池塘在投放消毒剂后的扩散过程,综合考虑所需要的总扩散时间,以及所需消毒剂的总量,得出一种较为合理的投放方式。从而,为池塘养殖提供一定的参考。
[Abstract]:Convection-diffusion equations play an important role in the field of environment, such as the distribution of pollutants in rivers and atmosphere, which can be described by convection-diffusion equations. Therefore, it is of great significance to study the solution of convection-diffusion equations. At present, the finite difference method is one of the most important numerical methods in the process of solving the convection-diffusion equation. However, the stability of explicit schemes in this method is limited, and implicit schemes are difficult to parallel because they need to solve large linear equations. In recent years, with the rapid development of high-performance computers and the rapid development of parallel algorithms, researchers at home and abroad have made a lot of research results. For example, Evans and Zhang Baolin proposed a series of parallel algorithms for diffusion equation and convection diffusion equation. However, some of the parallel algorithms have low accuracy, so it is necessary to construct high precision parallel algorithms to improve the speed and accuracy of computation. On the basis of implicit scheme, this paper constructs a new parallel algorithm for one-dimensional convection-diffusion equation of concentration and two-dimensional convection-diffusion equation of concentration, and applies the high-precision parallel algorithm of two-dimensional concentration diffusion equation to the practical problem of environment. In the second chapter of this paper, a new class of second-order precision implicit parallel schemes is constructed for the one-dimensional, two-dimensional, one-dimensional, two-dimensional convection-diffusion equations of concentration. The calculation results are analyzed and the validity of the scheme is verified. In the third chapter of this paper, a new high-order precision parallel scheme is constructed for the one-dimensional concentration diffusion equation, the one-dimensional concentration convection-diffusion equation and the two-dimensional concentration diffusion equation. By comparing the numerical solution with the exact solution, it is found that the curves of the two solutions are in perfect agreement with each other, and the calculation accuracy is obviously higher than that of the second-order precision implicit parallel scheme, which proves the validity of the scheme. In the fourth chapter of this paper, based on the high precision parallel scheme of two-dimensional concentration diffusion equation, the diffusion process of aquaculture pond after disinfectant is studied, and the total diffusion time is considered synthetically. And the total amount of disinfectants needed to get a more reasonable way of delivery. Thus, to provide a certain reference for pond culture.
【学位授予单位】：大连海事大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

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