Comparison Research On The Parallel Algorithm For Solving High Dimensional Wave Equations

Acoustics is a classic subject, which is widely applied in aviation, vehicles, ships, machinery manufacturing. In the acoustic filed, the wave equation is used to describe changes. However, rigorously analytical solution can hardly be obtained by this method. The calculation of numerical solutions is of great significance. There are problems of large amount of data, large memory, and low operational efficiency during the process of solving the numerical solution. To obtain the numerical solution of wave equation precisely and efficiently, a finite difference method is used to simulate the high-dimensional wave equation. Under the premise of ensuring the accuracy of numerical solution, two parallel iterative algorithms are constructed. Numerical accuracy and convergence speed and other aspects of the algorithm are analyzed by numerical examples of two-dimensional and three-dimensional wave equations.For general wave equation, comparison of local truncation error and stability analysis is performed according to local truncation error including explicit, implicit, alternating direction format. Although the explicit format is easy for direct calculation and has an excellent parallelism, it is constrained by the stability. Furthermore, the more dimensions, the more severe the conditions. The implicit format is unconditionally stable, it also need to solve large linear equations with wide band coefficient matrix. This paper focuses on the alternating direction scheme with advantages of above both formats.Alternating direction implicit scheme of two or three-dimensional is analyzed. The implicit difference scheme is calculated in one direction each time. On this base, two iterative parallel algorithms of different formats are emphatically contrasted. The core idea is to split the large linear equations into several sub-equations solved simultaneously. These equations are calculated independently and are interrelated during the iterative process. Computer resources are sufficiently used, and the solving efficiency is improved. For high-dimensional equations with many computing nodes, such parallel algorithms have obvious advantages.Two parallel iterative algorithms are applied to different types of numerical example in the software platform of MATLAB. The numerical results are compared with the theoretical results. It is shown that the errors between numerical and analytical solution are in the allowable range. Both algorithms reflect good parallelism, and the high-dimensional problem is simplistic. A large sub-matrix order of the coefficient matrix as well as a small ratio of time step and spatial step is beneficial to a rapid convergence of the algorithm. Different estimates of the first time horizon as well as the ratio of time step and spatial step has different influence on the error precision. Under the same conditions, the convergence rate of the second iterative parallel algorithms is nearly doubled compared with the first algorithm.