高维波动方程几种并行算法的比较分析研究

发布时间：2018-01-26 01:41

本文关键词： 高维波动方程有限差分交替方向隐式格式并行算法　出处：《大连理工大学》2015年硕士论文　论文类型：学位论文

【摘要】：声学是一门在航海航空、车辆船舶、机械制造等多领域广泛应用的经典学科。在声学领域,人们采用波动方程来描述声场分布变化。但是面对实际工程领域问题,波动方程很少能得到严格的理论解析解,所以数值解的计算有着十分重要的意义。同时用计算机求解数值解的过程中,存在着数据量大,占用内存大,运算效率低的问题。为了准确、有效快捷地求解双曲型波动方程的数值解,本文针对高维波动问题,对古典显格式和交替方向隐式(ADI)格式的两种并行算法进行了比较分析研究,通过二维和三维的数值算例对算法的数值精度和收敛速度等方面进行了分析。针对一般的波动方程,根据其显格式、隐格式、交替方向隐式(ADI)格式三种有限差分格式进行局部截断误差、稳定性分析对比,显格式虽然便于直接计算,有良好的并行性,但是其受稳定性条件约束,而且维数越多条件越苛刻：而隐格式通常是无条件稳定,但是要解系数矩阵为宽带状的大规模线性方程组。本文重点研究具备上述两种格式优点的交替方向隐式格式。本文分析了二维及三维波动方程的交替方向隐式格式,即每次只在一个方向进行隐式格式差分计算,形成了三对角线的线性方程组。并在此基础上着重对比了两种不同格式的迭代并行算法。这种算法核心内容是将大型线性方程组分裂为多个子方程组同时求解。这些子方程组分别计算时相互独立,迭代过程中又相互关联,充分利用计算机资源,提高了求解效率。针对计算节点多的高维方程,此类并行算法的优势更加明显。利用MATLAB软件编程,将两种迭代并行算法应用于不同数值算例,将数值计算结果与理论精确解对比,结果表明：数值解与解析解的误差在允许范围内,同时两种算法均体现出良好的并行性,将高维问题简单化；较大的系数矩阵子矩阵阶数、较小的网格比(时间步长与空间步长比)利于算法的快速收敛；不同的初始时刻计算值,网格比对误差精度影响不同；系数矩阵子矩阵阶数越大,计算时间越短；同等条件下,第二种迭代并行算法较第一种算法的收敛速度提高了近一倍。
[Abstract]:Acoustics is a classical subject which is widely used in many fields, such as navigation aviation, vehicle and ship, machinery manufacture, etc. It is in the field of acoustics. The wave equation is used to describe the variation of sound field distribution, but in the practical engineering field, the wave equation rarely gets a strict theoretical analytical solution. Therefore, the calculation of numerical solution is very important. In the process of solving numerical solution by computer, there are many problems, such as large amount of data, large amount of memory and low efficiency. The numerical solution of hyperbolic wave equation is solved efficiently and quickly. In this paper, two parallel algorithms of classical explicit scheme and alternating direction implicit scheme are compared and studied for the high dimensional wave problem. The numerical accuracy and convergence rate of the algorithm are analyzed by two and three dimensional numerical examples. According to the explicit scheme and implicit scheme for the general wave equation. The local truncation error of the three finite difference schemes is analyzed and compared. The explicit scheme is easy to calculate directly and has good parallelism, but it is constrained by the stability condition. And the more dimension the more stringent the conditions: implicit schemes are usually unconditionally stable. However, in order to solve large scale linear equations with wideband coefficient matrix, this paper focuses on the alternating direction implicit schemes with the advantages of the above two schemes. In this paper, the alternating direction implicit lattices of two-dimensional and three-dimensional wave equations are analyzed. Style. That is to say, the implicit scheme difference calculation is only carried out in one direction at a time. A tridiagonal system of linear equations is formed. On this basis, two iterative parallel algorithms with different schemes are compared. The core of the algorithm is to divide the large linear equations into multiple subequations and solve them simultaneously. These subequations are independent of each other when they are calculated separately. The iterative process is related to each other, making full use of computer resources to improve the efficiency of the solution. For high-dimensional equations with more nodes, the advantages of this kind of parallel algorithm are more obvious. MATLAB software is used to program. Two iterative parallel algorithms are applied to different numerical examples. The numerical results are compared with the theoretical exact solutions. The results show that the error between the numerical solution and the analytical solution is within the allowable range. At the same time, the two algorithms show good parallelism and simplify the high-dimensional problem. Larger order of coefficient matrix submatrix and smaller mesh ratio (time step to space step ratio) are beneficial to fast convergence of the algorithm. At different initial time, the error accuracy of grid ratio is different. The larger the order of coefficient matrix is, the shorter the calculation time is. Under the same conditions, the convergence speed of the second iterative parallel algorithm is nearly double that of the first.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O246

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