本文選題：變系數(shù)Helmholtz方程 + 有限差分法�。� 參考：《寧夏大學(xué)》2017年碩士論文

【摘要】：生活中的很多物理現(xiàn)象都可以用Helmholtz方程來刻畫,例如時諧波的傳播、水下聲學(xué)、航空聲學(xué)、電磁波散射等.近年來國內(nèi)外學(xué)者采用有限元法、有限體積法、有限差分法等數(shù)值方法在Helmholtz方程的求解方面做了大量的研究.在實際生活中,由于某些物理屬性的變化,或是在非均勻介質(zhì)中時諧波的傳播問題,需要求解變系數(shù)Helmholtz方程.本文首先對二維變系數(shù)Helmholtz方程做簡單的變形,利用中心差分離散原方程得到二階緊致差分格式,對二階格式的截斷誤差項進(jìn)行修正,從而得到求解原方程的四階緊致差分格式,同時對局部Sommerfeld-type邊界條件的四階精度的差分格式進(jìn)行逼近.其次將二維格式推廣到三維,得到三維變系數(shù)Helmholtz方程的四階緊致差分格式.最后利用GMRES(m)迭代法對帶有Dirichlet邊界問題和局部Sommerfeld-type邊界問題進(jìn)行數(shù)值驗證,結(jié)果表明了格式的有效性和可行性,同時通過對直接法和GMRES(m)方法的計算時間比較,結(jié)果表明在三維問題的求解上,GMRES(m)算法有很大的優(yōu)勢.
[Abstract]:Many physical phenomena in life can be described by Helmholtz equation, such as time-harmonic propagation, underwater acoustics, aero-acoustics, electromagnetic wave scattering and so on. In recent years, many numerical methods, such as finite element method, finite volume method and finite difference method, have been used to solve Helmholtz equations at home and abroad. In real life, due to the change of some physical properties or the problem of harmonic propagation in inhomogeneous media, the variable coefficient Helmholtz equation needs to be solved. In this paper, the two-dimensional Helmholtz equation with variable coefficients is simply deformed, the second order compact difference scheme is obtained by using the central difference discrete original equation, the truncation error term of the second order scheme is corrected, and the fourth-order compact difference scheme for solving the original equation is obtained. At the same time, the fourth order difference scheme of local Sommerfeld-type boundary condition is approximated. Secondly, the two-dimensional scheme is extended to 3D, and the four-order compact difference scheme of three-dimensional Helmholtz equation with variable coefficients is obtained. Finally, the numerical results of the Dirichlet boundary problem and the local Sommerfeld-type boundary problem are verified by using the GMRESm) iteration method. The results show that the scheme is effective and feasible. At the same time, the computational time of the direct method and the GM method is compared. The results show that the GMRESm) algorithm has a great advantage in solving 3D problems.
【學(xué)位授予單位】：寧夏大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.8

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