变系数Helmholtz方程迭代方法研究

发布时间：2018-05-20 10:29

本文选题：变系数Helmholtz方程 + 有限差分法　；参考：《宁夏大学》2017年硕士论文

【摘要】：生活中的很多物理现象都可以用Helmholtz方程来刻画,例如时谐波的传播、水下声学、航空声学、电磁波散射等.近年来国内外学者采用有限元法、有限体积法、有限差分法等数值方法在Helmholtz方程的求解方面做了大量的研究.在实际生活中,由于某些物理属性的变化,或是在非均匀介质中时谐波的传播问题,需要求解变系数Helmholtz方程.本文首先对二维变系数Helmholtz方程做简单的变形,利用中心差分离散原方程得到二阶紧致差分格式,对二阶格式的截断误差项进行修正,从而得到求解原方程的四阶紧致差分格式,同时对局部Sommerfeld-type边界条件的四阶精度的差分格式进行逼近.其次将二维格式推广到三维,得到三维变系数Helmholtz方程的四阶紧致差分格式.最后利用GMRES(m)迭代法对带有Dirichlet边界问题和局部Sommerfeld-type边界问题进行数值验证,结果表明了格式的有效性和可行性,同时通过对直接法和GMRES(m)方法的计算时间比较,结果表明在三维问题的求解上,GMRES(m)算法有很大的优势.
[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.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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