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 计算数学 2018, Vol. 40 Issue (2): 171-190    DOI:
 论文 最新目录 | 下期目录 | 过刊浏览 | 高级检索 Previous Articles  |  Next Articles Helmholtz方程有限差分方法概述

1. 重庆大学数学与统计学院, 重庆 401331;
2. 石河子大学理学院数学系, 石河子 832003
FINITE DIFFERENCE METHODS FOR THE HELMHOLTZ EQUATION: A BRIEF REVIEW
Wang Kun1, Zhang Yang1, Guo Rui2
1. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China;
2. Department of Mathematics, Faculty of Sciences, Shihezi University, Shihezi 832003, China
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Abstract： In this paper, a brief review on the development of the finite difference methods for the Helmholtz equation in the latest twenty years is presented. Based on the phase error, the results in 1D, 2D and 3D are given in this paper, the difference and relationship of different schemes are also shown, especially on the computational effect when applying to solve the Helmholtz equation with high wave numbers, moreover, some mainly problems existing in approximating the high wave number problems are discussed.

 引用本文: . Helmholtz方程有限差分方法概述[J]. 计算数学, 2018, 40(2): 171-190. . FINITE DIFFERENCE METHODS FOR THE HELMHOLTZ EQUATION: A BRIEF REVIEW[J]. Mathematica Numerica Sinica, 2018, 40(2): 171-190.

  Ihlenburg F. Finite Element Analysis of Acoustic Scattering[M]. Spring, NewYork, 1998.  Bao G, Sun W. A fast algorithm for the electromaginetic scattering from a large cavity[J]. SIAM J. Sci. Comput., 2005, 27:553-574. Bao G, Yun K, Zhou Z. Stability of the scattering from a large electromagnetic cavity in two dimensions[J]. SIAM J. Math. Anal., 2012, 44:383-404. Chen Z, Liu X. An adaptive perfectly matched layer technique for time-harmonic scattering problems[J]. SIAM J. Numer. Anal., 2005, 43(2):645-671. Chen Z, Liang C, Xiang X. An anisotropic perfectly matched layer method for Helmholtz scattering problems with discontinuous wave number[J]. Inverse Probl. Imag., 2013, 7:663-678. Ma J, Zhu J, Li M. The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber[J]. Engrg. Anal. Bound. Elem., 2010, 34:1058-1063. Hsiao G, Liu F, Sun J, Xu L. A coupled BEM and FEM for the interior transmission problem in acoustics[J]. J. Comput. Appl. Math., 2011, 235(17):5213-5221. Hsiao G, Xu L. A system of boundary integral equations for the transmission problem in acoustics[J]. Appl. Numer. Math., 2011, 61(9):1017-1029. Wang Y, Ma F, Zheng E. Galerkin method for the scattering problem of a slit[J]. J. Sci. Comput., 2017, 70(1):192-209. Wu H, Liu Y, Jiang W. Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems[J]. Eng. Anal. Bound. Elem., 2012, 63(2):248-254.  Geng H, Yin T, Xu L. A priori error estimates of the DtN-FEM for the transmission problem in acoustics[J]. J. Comput. Appl. Math., 2017, 313:1-17.  Sun W, Ma F. An error estimate of the coupled finite-infinite element method for scattering from an arc[J]. Int. J. Numer. Anal. Model., 2014, 11(4):841-853.  Shen J, Wang L. Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains[J]. SIAM J. Numer. Anal., 2007, 45:1954-1978. Shen J, Wang L. Spectral approximation of the Helmholtz equation with high wave numbers[J]. SIAM J. Numer. Anal., 2005, 43:623-644. Baruch G, Fibich G, Tsynkov S, Turkel E. Fourth order schemes for time-harmonic wave equations with discontinuous coefficients[J]. Commun. Comput. Phys., 2009, 5(2-4):442-455.  Britt S, Tsynkov S, Turkel E. A compact fourth order scheme for the Helmholtz equation in polar coordinates[J]. J. Sci. Comput., 2010, 54:26-47.  Britt S, Tsynkov S, Turkel E. Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes[J]. Commun. Comput. Phys., 2011, 9(3):520-541. Chen Z, Wu T, Yang H. An optimal 25-point finite difference scheme for the Helmholtz equation with PML[J]. J. Comput. Appl. Math., 2011, 236(6):1240-1258. Chen Z, Cheng D, Wu T. A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation[J]. J. Comput. Phys., 2012, 231(24):8152-8175. Chen Z, Cheng D, Feng W, Wu T. An optimal 9-point finite difference scheme for the Helmholtz equation with PML[J]. Int. J. Numer. Anal. Model., 2013, 10:389-410.  Cheng D, Liu Z, Wu T. A multigrid-based preconditioned solver for the Helmholtz equation with a discretization by 25-point difference scheme[J]. Math. Comput. Simul., 2015, 117:54-67. Cheng D, Tan X, Zeng T. A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting[J]. Comput. Math. Appl., 2017, 73(11):2345-2359. Fernandes D. A. Loula, Quasi optimal finite difference method for Helmholtz problem on unstructured grids[J]. Int. J. Numer. Meth. Engng., 2010, 82:1244-1281.  Feng X, Li Z, Qiao Z. High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients[J]. J. Comput. Math., 2011, 29:324-340. Feng X. A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient[J]. Int. J. Comput. Math., 2012, 89(5):618-624. Fu Y. Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers[J]. J. Comput. Math., 2008, 26:98-111.  Guo R, Wang K, Xu L. Efficient finite difference methods for acoustic scattering from circular cylindrical obstacle[J]. Int. J. Numer. Anal. Model., 2016, 13:986-1002.  Harari I, Turkel E. Accurate finite difference methods for time-harmonic wave propagation[J]. J. Comput. Phys., 1995, 119:252-270. Jo C, Shin C, Suh J. An optimal 9-point, finite-difference, frequency-space 2-D scalar wave extrapolator[J]. Geophysics, 1996, 61:529-537. Lambe L, Luczak R, Nehrbass J. A new finite difference method for the Helmholtz equation using symbolic computation[J]. Int. J. Comput. Engrg. Sci., 2003, 4:121-144. 刘佳勇. 二维大波数Helmholtz方程的九点差分格式[D]. 重庆大学, 2015.  Nabavia M, Siddiqui M, Dargahi J. A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation[J]. J. Sound Vibrat., 2007, 307:972-982. Operto S, Virieux J, Amestoy P, L'Excellent J, Giraud L, Ali H. 3D finite-difference frequencydomain modeling of visco-acoustic wave propagation using a massively parallel direct solver:A feasibility study[J]. Geophysics, 2007, 72:SM195-SM211.  Shin C, Sohn H. A frequency-space 2-D scalar wave extrapolator using extended 25-point finitedifference operator[J]. Geophysics, 1998, 63:289-296. Singer I, Turkel E. Sixth-order accurate finite difference schemes for the Helmholtz equation[J]. J. Comput. Acoust., 2006, 14:339-351. Singer I, Turkel E. High-order finite difference methods for the Helmholtz equation[J]. Comput. Methods Appl. Mech. Engrg., 1998, 163:343-358. Sutmann G. Compact finite difference schemes of sixth order for the Helmholtz equation[J]. J. Comput. Appl. Math., 2007, 203:15-31. Su X, Feng X, Li Z. Fourth-order compact schemes for Helmholtz equations with piecewise wave numbers in the polar coordinates[J]. J. Comput. Math., 2016, 34(5):499-510. Stephane O, Jean V, Patrick A, Jean-Yves E, Giraud G, Hafedh B. 3D finite-difference frequencydomain modeling of visco-acoustic wave propagation using a massively parallel direct solver:A feasibility study[J]. Geophysics, 2007, 72:SM195-SM211.  Turkel E, Gordon D, Gordon R, Tsynkov S. Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number[J]. J. Comput. Phys., 2013, 232:272-287. Wang K, Wong Y S. Pollution-free finite difference schemes for non-homogeneous Helmholtz equation[J]. Int. J. Numer. Anal. Model., 2014, 11:787-815.  Wang K, Wong Y S, Deng J. Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates[J]. Commun. Comput. Phys., 2015, 17:779-807. Wang K, Wong Y S. Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?[J]. Commun. Comput. Phys., 2017, 21:490-514. Wang K, Wong Y, Huang J. Solving Helmholtz equation at high wave numbers in exterior domains[J]. Appl. Math. Comput., 2017, 298:221-235.  Wang K, Wong Y S, Huang J. Analysis of pollution-free approaches for multi-dimensional Helmholtz equations[J]. submitted.  Wong Y S, Li G. Exact finite difference schemes for solving Helmholtz equation at any wavenumber[J]. Int. J. Numer. Anal. Model. Ser., B 2011, 2:91-108.  Wu T, Chen Z. A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML[J]. J. Comput. Appl. Math., 2014, 267:82-95. Wu T, Chen Z, Chen J. Optimal 25-point finite-difference subgridding techniques for the 2D Helmholtz equation[J]. Math. Probl. Eng., 2016, Art. ID 1719846, 16 pp.  Wu T. A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equation[J]. J. Comput. Appl. Math., 2017, 311:497-512. Zhang W, Dai Y. Finite-difference solution of the Helmholtz equation based on two domain decomposition algorithms[J]. J. Appl. Math. Phys., 2013, 1:18-24.  Babuaska I, Sauter S. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers[J]. SIAM J. Numer. Anal., 1997, 34(6):2392-2423. Burman E, Wu H, Zhu L. Linear continuous interior penalty finite element method for Helmholtz equation with high wave number:one-dimensional analysis[J]. Numer Methods Partial Differential Eq., 2016, 32:1378-1410. Chen H, Lu P, Xu X. A hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2013, 51(4):2166-2188. Chen H, Lu P, Xu X. A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation[J]. J. Comput. Phys., 2014, 264:133-151. Du Y, Wu H. Preasymptotic error analysis of higher order FEM and CIP-FEM for Helmholtz equation with high wave number[J]. SIAM J. Numer. Anal., 2015, 53(2):782-804. Feng X, Wu H. hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number[J]. Math. Comp., 2011, 80(276):1997-2024. Chen W, Liu Y, Xu X. A robust domain decomposition method for the Helmholtz equation with high wave number[J]. ESAIM Math. Model. Numer. Anal., 2016, 50(3):921-944. Harari I, Hughes T J R. Finite element method for the Helmholtz equation in an exterior domain:model problems[J]. Comp. Methods Appl. Mech. Eng., 1991, 87:59-96. Wu H. Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I:linear version[J]. IMA J. Numer. Anal., 2014, 34(3):1266-1288. Zhu L, Wu H. Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part Ⅱ:hp version[J]. SIAM J. Numer. Anal., 2013, 51(3):1828-1852. Zheng E, Ma F, Wang Y. A least-squares FEM for the direct and inverse rectangular cavity scattering problem[J], Math. Probl. Eng., 2015, Art. ID 524345, 10 pp.  Han H, Huang Z. A tailored finite point method for the Helmholtz equation with high wave numbers in heterogenous medium[J]. J. Comput. Math., 2008, 26(5):728-739.  Huang Z, Yang X. Tailored finite cell method for solving Helmholtz equation in layered Heterogeneous medium[J]. J. Comput. Math., 2012, 30:381-391. Hu Q, Yuan L. A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations[J]. Int. J. Numer. Anal. Model., 2014, 11(3):587-607.  Li S, Xiang S. Convergence analysis of a coupled method for Helmholtz equation[J]. Complex Var. Elliptic Equ., 2014, 59(4):484-503. Zheng C. Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation[J]. Commun. Math. Sci., 2010, 8:1041-1066. Zheng C. Global geometrical optics approximation to the high frequency Helmholtz equation with discontinuous media[J]. Commun. Math. Sci., 2015, 13:1949-1974. Chen Z, Cheng D, Feng W, Wu T, Yang H. A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML[J]. J. Math. Anal. Appl., 2011, 383(2):522-540. Chen H, Wu H, Xu X. Multilevel preconditioner with stable coarse grid corrections for the Helmholtz equation[J]. SIAM J. Sci. Comput., 2015, 37(1):A221-A244.  Ito K, Qiao Z, Toivanen J. A domain decomposition solver for acoustic scattering by elastic objects in layered media[J]. J. Comput. Phys., 2008, 227(19):8685-8698. Yuan L, Hu Q, An H. Parallel preconditioners for plane wave Helmholtz and Maxwell systems with large wave numbers[J]. Int. J. Numer. Anal. Model., 2016, 13(5):802-819.  Engquist B, Ying L. Sweeping preconditioner for the Helmholtz equation:Hierarchical matrix representation[J]. Commun. Pure Appl. Math., 2011, LXIV:697-735.  Hu Q, Zhang H. Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations[J]. SIAM J. Sci. Comput., 2016, 38(4):A2232-A2261.  Yuan L, Hu Q. A solver for Helmholtz system generated by the discretization of wave shape functions[J]. Adv. Appl. Math. Mech., 2013, 5(6):791-808. Li S, Xiang S, Xian J. A fast hybrid Galerkin method for high-frequency acoustic scattering[J]. Appl. Anal., 2017, 96(10):1698-1712. Zhao M, Qiao Z, Tang T. A fast high order method for electromagnetic scattering by large open cavities[J]. J. Comput. Math., 2011, 29(3):287-304.
  付姚姚, 曹礼群. 矩阵形式二次修正Maxwell-Dirac系统的多尺度算法[J]. 计算数学, 2019, 41(4): 419-439.  尹旭, 卢朓, 姜海燕. 数值求解含时Wigner方程的一种高阶算法[J]. 计算数学, 2019, 40(1): 21-33.  澈力木格, 何斯日古楞, 李宏. 大气污染模型的POD基降维有限差分算法[J]. 计算数学, 2018, 39(3): 172-182.  卢培培, 许学军. 高波数波动问题的多水平方法[J]. 计算数学, 2018, 40(2): 119-134.  武海军. 高波数Helmholtz方程的有限元方法和连续内罚有限元方法[J]. 计算数学, 2018, 40(2): 191-213.  骆其伦, 黎稳. 二维Helmholtz方程的联合紧致差分离散方程组的预处理方法[J]. 计算数学, 2017, 39(4): 407-420.  郑权, 高玥, 秦凤. Helmholtz方程外边值问题的基于修正的DtN边界条件的有限元方法[J]. 计算数学, 2016, 38(2): 200-211.  袁龙, 胡齐芽. 复波数Helmholtz方程和时谐Maxwell方程组的平面波间断Petrov-Galerkin方法[J]. 计算数学, 2015, 36(3): 185-196.  孟文辉, 王连堂. Helmholtz方程周期Green函数及其偏导数截断误差收敛阶的分析[J]. 计算数学, 2015, 37(2): 123-136.  陈璐, 王雨顺. 保结构算法的相位误差分析及其修正[J]. 计算数学, 2014, 36(3): 271-290.  柯日焕, 黎稳. 用CCD法离散求解二维Helmholtz方程的数值方法[J]. 计算数学, 2013, 34(3): 221-230.  段艳婷, 王连堂, 徐建丽. 二维Helmholtz方程外问题的数值解法[J]. 计算数学, 2011, 32(1): 57-63.  张敏,杜其奎,. 椭圆外区域上Helmholtz问题的自然边界元法[J]. 计算数学, 2008, 30(1): 75-88.  杨超,孙家昶. 一类六边形网格上拉普拉斯4点差分格式及其预条件子[J]. 计算数学, 2005, 27(4): 437-448.  贾祖朋,邬吉明,余德浩. 三维Helmholtz方程外问题的自然边界元与有限元耦合法[J]. 计算数学, 2001, 23(3): 357-368.
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