An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation

WANG Zhaojun; WU Tingting; ZENG Taishan

doi:10.6054/j.jscnun.2022031

Journal of South China Normal University (Natural Science Edition) > 2022 > 54(2): 90-100. > DOI: 10.6054/j.jscnun.2022031

WANG Zhaojun, WU Tingting, ZENG Taishan. An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation[J]. Journal of South China Normal University (Natural Science Edition), 2022, 54(2): 90-100. DOI: 10.6054/j.jscnun.2022031

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An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
2.
School of Mathematics, South China Normal University, Guangzhou 510631, China
3.
Guangdong Key Laboratory of Big Data Analysis and Processing, Guangzhou 510006, China

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Received Date: September 01, 2021
Available Online: May 11, 2022

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Abstract

Abstract

To obtain a finite difference scheme with high accuracy for solving the 2D Helmholtz equation, an improved compact sixth-order difference scheme is constructed. Firstly, the truncation error of a compact sixth-order difference scheme with optimal parameters is presented. Then, some terms of the truncation error are approximated with second-order compact formulas to obtain an improved compact difference scheme. Next, the convergence analysis of the improved compact difference scheme is given, and it is proved that the proposed scheme enjoys sixth-order convergence. Based on minimizing the numerical dispersion, a refined choice strategy is proposed for choosing weight parameters. Compared with the compact sixth-order difference scheme with optimal parameters, numerical experiments show that, the numerical accuracy of the improved compact sixth-order difference scheme has been significantly improved, and the scheme's error is less dependent on the wavenumber k.
- Helmholtz equation,
- compact finite difference scheme,
- numerical dispersion

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[1]	BERENGER J P. A perfectly matched layer for the absorption of electromagnetic waves[J]. Journal of Computational Physics, 1994, 114(2): 185-200. doi: 10.1006/jcph.1994.1159
[2]	TEZAUR R, MACEDO A, FARHAT C, et al. Three-dimensional finite element calculations in acoustic scatte-ring using arbitrarily shaped convex artificial boundaries[J]. International Journal for Numerical Methods in Engineering, 2002, 53(6): 1461-1476. doi: 10.1002/nme.346
[3]	JO C H, SHIN C, SUH J H. An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator[J]. Geophysics, 1996, 61(2): 529-537. doi: 10.1190/1.1443979
[4]	王坤, 张扬, 郭瑞. Helmholtz方程有限差分方法概述[J]. 计算数学, 2018, 40(2): 171-190. https://www.cnki.com.cn/Article/CJFDTOTAL-JSSX201802005.htm WANG K, ZHANG Y, GUO R. Finite difference methods for the Helmholtz equation: a brief review[J]. Mathema-tica Numerica Sinica, 2018, 40(2): 171-190. https://www.cnki.com.cn/Article/CJFDTOTAL-JSSX201802005.htm
[5]	BABUŠKA I, SAUTER S A. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?[J]. SIAM Journal on Numerical Analysis, 1997, 34(6): 2392-2423. doi: 10.1137/S0036142994269186
[6]	BABUŠKA I, IHLENBURG F, PAIK E T, et al. A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 128(3/4): 325-359.
[7]	IHLENBURG F, BABUŠKA I. Finite element solution of the Helmholtz equation with high wave number, Part I: the h-version of the FEM[J]. Computers and Mathematics with Applications, 1995, 30(9): 9-37. doi: 10.1016/0898-1221(95)00144-N
[8]	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. https://www.cnki.com.cn/Article/CJFDTOTAL-JSSX201802006.htm WU H J. FEM and CIP-FEM for Helmholtz equation with high wave number[J]. Mathematica Numerica Sinica, 2018, 40(2): 191-213. https://www.cnki.com.cn/Article/CJFDTOTAL-JSSX201802006.htm
[9]	CHENG D S, TAN X, ZENG T S. A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting[J]. Computers and Mathematics with Applications, 2017, 73(11): 2345-2359. doi: 10.1016/j.camwa.2017.04.005
[10]	FU Y P. Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers[J]. Journal of Computational Mathematics, 2008, 26(1): 98-111.
[11]	HARARI I, TURKEL E. Accurate finite difference methods for time-harmonic wave propagation[J]. Journal of Computational Physics, 1995, 119: 252-270. doi: 10.1006/jcph.1995.1134
[12]	KUMAR N, DUBEY R K. A new development of sixth order accurate compact scheme for the Helmholtz equation[J]. Journal of Applied Mathematics and Computing, 2020, 62: 637-662. doi: 10.1007/s12190-019-01301-x
[13]	SINGER I, TURKEL E. High-order finite difference methods for the Helmholtz equation[J]. Computer Methods in Applied Mechanics and Engineering, 1998, 163(1/2/3/4): 343-358.
[14]	SUTMANN G. Compact finite difference schemes of sixth order for the Helmholtz equation[J]. Journal of Computational and Applied Mathematics, 2007, 203(1): 15-31. doi: 10.1016/j.cam.2006.03.008
[15]	WU T T. A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equation[J]. Journal of Computational and Applied Mathematics, 2017, 311: 497-512. doi: 10.1016/j.cam.2016.08.018
[16]	WU T T, XU R M. An optimal compact sixth-order finite difference scheme for the Helmholtz equation[J]. Computers and Mathematics with Applications, 2018, 75(7) : 2520- 2537. doi: 10.1016/j.camwa.2017.12.023
[17]	朱昌允, 秦国良, 徐忠. Chebyshev谱元方法结合并行算法求解三维区域的Helmholtz方程[J]. 应用力学学报, 2012, 29(3): 247-251. https://www.cnki.com.cn/Article/CJFDTOTAL-YYLX201203005.htm ZHU C Y, QIN G L, XU Z. Parallel Chebyshev spectral element method for Helmholtz equation in 3D domain[J]. Chinese Journal of Applied Mechanics, 2012, 29(3): 247-251. https://www.cnki.com.cn/Article/CJFDTOTAL-YYLX201203005.htm
[18]	程东升, 刘志勇, 薛国伟, 等. 一种针对大波数Helmholtz方程的高性能并行预条件迭代求解算法[J]. 计算机科学, 2018, 45(7): 299-306. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJA201807051.htm CHENG D S, LIU Z Y, XUE G W, et al. High-perfor-mance parallel preconditioned iterative solver for Helmholtz equation with large wavenumbers[J]. Computer Science, 2018, 45(7): 299-306. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJA201807051.htm
[19]	WU T T, SUN Y R, CHENG D S. A new finite difference scheme for the 3D Helmholtz equation with a preconditioned iterative solver[J]. Applied Numerical Mathema-tics, 2021, 161: 348-371. doi: 10.1016/j.apnum.2020.11.023
[20]	CHEN Z Y, CHENG D S, FENG W, et al. An optimal 9-point finite difference scheme for the Helmholtz equation with PML[J]. International Journal of Numerical Analysis and Modeling, 2013, 10(2): 389-410.
[21]	TUEKEL E, GORDON D, GORDON R, et al. Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number[J]. Journal of Computational Physics, 2013, 232(1): 272-287.
[22]	杨胜良. 三对角行列式及其应用[J]. 工科数学, 2002, 18(2): 102-104. https://www.cnki.com.cn/Article/CJFDTOTAL-GKSX200202025.htm YANG S L. Tridiagonal determinant and its applications[J]. Journal of Mathematics for Technology, 2002, 18(2): 102-104. https://www.cnki.com.cn/Article/CJFDTOTAL-GKSX200202025.htm
[23]	ERLANGGA Y, TURKEL E. Iterative schemes for high order compact discretizations to the exterior Helmholtz equation[J]. ESAIM: Mathematical Modelling and Nume-rical Analysis, 2012, 46(3): 647-660.

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