本文關(guān)鍵詞： 高維波動(dòng)方程 有限差分 交替方向隱式格式 并行算法　出處：《大連理工大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：聲學(xué)是一門在航海航空、車輛船舶、機(jī)械制造等多領(lǐng)域廣泛應(yīng)用的經(jīng)典學(xué)科。在聲學(xué)領(lǐng)域,人們采用波動(dòng)方程來描述聲場(chǎng)分布變化。但是面對(duì)實(shí)際工程領(lǐng)域問題,波動(dòng)方程很少能得到嚴(yán)格的理論解析解,所以數(shù)值解的計(jì)算有著十分重要的意義。同時(shí)用計(jì)算機(jī)求解數(shù)值解的過程中,存在著數(shù)據(jù)量大,占用內(nèi)存大,運(yùn)算效率低的問題。為了準(zhǔn)確、有效快捷地求解雙曲型波動(dòng)方程的數(shù)值解,本文針對(duì)高維波動(dòng)問題,對(duì)古典顯格式和交替方向隱式(ADI)格式的兩種并行算法進(jìn)行了比較分析研究,通過二維和三維的數(shù)值算例對(duì)算法的數(shù)值精度和收斂速度等方面進(jìn)行了分析。針對(duì)一般的波動(dòng)方程,根據(jù)其顯格式、隱格式、交替方向隱式(ADI)格式三種有限差分格式進(jìn)行局部截?cái)嗾`差、穩(wěn)定性分析對(duì)比,顯格式雖然便于直接計(jì)算,有良好的并行性,但是其受穩(wěn)定性條件約束,而且維數(shù)越多條件越苛刻：而隱格式通常是無條件穩(wěn)定,但是要解系數(shù)矩陣為寬帶狀的大規(guī)模線性方程組。本文重點(diǎn)研究具備上述兩種格式優(yōu)點(diǎn)的交替方向隱式格式。本文分析了二維及三維波動(dòng)方程的交替方向隱式格式,即每次只在一個(gè)方向進(jìn)行隱式格式差分計(jì)算,形成了三對(duì)角線的線性方程組。并在此基礎(chǔ)上著重對(duì)比了兩種不同格式的迭代并行算法。這種算法核心內(nèi)容是將大型線性方程組分裂為多個(gè)子方程組同時(shí)求解。這些子方程組分別計(jì)算時(shí)相互獨(dú)立,迭代過程中又相互關(guān)聯(lián),充分利用計(jì)算機(jī)資源,提高了求解效率。針對(duì)計(jì)算節(jié)點(diǎn)多的高維方程,此類并行算法的優(yōu)勢(shì)更加明顯。利用MATLAB軟件編程,將兩種迭代并行算法應(yīng)用于不同數(shù)值算例,將數(shù)值計(jì)算結(jié)果與理論精確解對(duì)比,結(jié)果表明：數(shù)值解與解析解的誤差在允許范圍內(nèi),同時(shí)兩種算法均體現(xiàn)出良好的并行性,將高維問題簡(jiǎn)單化；較大的系數(shù)矩陣子矩陣階數(shù)、較小的網(wǎng)格比(時(shí)間步長(zhǎng)與空間步長(zhǎng)比)利于算法的快速收斂；不同的初始時(shí)刻計(jì)算值,網(wǎng)格比對(duì)誤差精度影響不同；系數(shù)矩陣子矩陣階數(shù)越大,計(jì)算時(shí)間越短；同等條件下,第二種迭代并行算法較第一種算法的收斂速度提高了近一倍。
[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.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O246

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本文編號(hào)：1464291