Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T09:28:44.347Z Has data issue: false hasContentIssue false
  • English
  • Français

Is Pollution Effect of Finite Difference Schemes Avoidable for Multi-Dimensional Helmholtz Equations with High Wave Numbers?

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  07 February 2017

Kun Wang  and
Yau Shu Wong
Kun Wang*
Affiliation:
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, P.R. China Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada Institute of Computing and Data Sciences, Chongqing University, Chongqing 400044, P.R. China
Yau Shu Wong*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
*
*Corresponding author.Email addresses:kunwang@cqu.edu.cn, wk580@163.com (K.Wang), yaushu.wong@ualberta.ca (Y. S. Wong)
*Corresponding author.Email addresses:kunwang@cqu.edu.cn, wk580@163.com (K.Wang), yaushu.wong@ualberta.ca (Y. S. Wong)
Get access

Abstract

This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

Keywords

Helmholtz equationseparation of variableshigh wave numberpollution free finite ifference schemelarge cavity problemrectangular waveguide

MSC classification

Secondary: 65N06: Finite difference methods 65N15: Error bounds 65N22: Solution of discretized equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 490 - 514
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ainsworth, M., Discrete dispersion relation for hp-version finite element approximation at high wave number, SIAM J. Numer. Anal. 42 (2005) 553–575.CrossRefGoogle Scholar
[2] Andouze, S., Goubet, O., Poullet, P., A multilevel method for solving the Helmholtz equation: the analysis of the one-dimensional case, Int. J. Numer. Anal. Model. 8 (2011) 365–372.Google Scholar
[3] Babus˘ka, I. Banerjee, U., Stable generalized finite element method (SGFEM), Comput. Methods Appl. Mech. Engrg. 201-204 (2012) 91–111.CrossRefGoogle Scholar
[4] Babus˘ka, I. Ihlenburg, F., Paik, E., Sauter, S., A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. Methods Appl. Mech. Engrg. 128 (1995) 325–359.CrossRefGoogle Scholar
[5] Babus˘ka, I. Sauter, S. A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Review 42 (2000) 451–484.Google Scholar
[6] Bao, G., Sun, W., A fast algorithm for the electromaginetic scattering from a large cavity, SIAM J. Sci. Comput. 27 (2005) 553–574.CrossRefGoogle Scholar
[7] Bao, G., Wei, G.W., Zhao, S., Numerical solution of the helmholtz equation with high wavenumbers, Int. J. Numer. Meth. Engng. 59 (2004) 389–408.CrossRefGoogle Scholar
[8] Bao, G., Yun, K., Zhou, Z., Stability of the scattering from a large electromagnetic cavity in two dimensions, SIAM J. Math. Anal. 44 (2012) 383–404.CrossRefGoogle Scholar
[9] Bollhofer, M., Grote, M., Schenk, O., Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media, SIAM J. Sci. Comput. 31 (2009) 3781–3805.CrossRefGoogle Scholar
[10] Britt, D., Tsynkov, S., Turkel, E., A higher-order numerical method for the Helmholtz equation with non-standard boundary conditions, submitted.Google Scholar
[11] Britt, S., Tsynkov, S., Turkel, E., Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes, Commun. Comput. Phys. 9 (2011) 520–541.CrossRefGoogle Scholar
[12] Britt, S., Tsynkov, S., Turkel, E., A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates, J. Sci. Comput. 45 (2010) 26–47.CrossRefGoogle Scholar
[13] Callihan, R. S., Wood, A. W., A modified Helmholtz equation with impedance boundary conditions, Adv. Appl. Math. Mech. 4 (2012) 703–718.CrossRefGoogle Scholar
[14] Chen, Z., Cheng, D., Feng, W., Wu, T., An optimal 9-point finite difference scheme for the Helmholtz equation with PML, Int. J. Numer. Anal. Model. 10 (2013) 389–410.Google Scholar
[15] Chen, Z., Cheng, D., Wu, T., A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation, J. Comput. Phys. 231 (2012) 8152–8175.CrossRefGoogle Scholar
[16] Chen, H., Lu, P., Xu, X., A robust multilevel method for hybridizable discontinuous Galerkin method for the Helmholtz equation, J. Comput. Phys. 264 (2014) 133–151.CrossRefGoogle Scholar
[17] Chen, H., Wu, H., Xu, X., Multilevel preconditioner with stable coarse grid corrections for the Helmholtz equation, SIAM J. Sci. Comput. 37 (2015) A221–A244.CrossRefGoogle Scholar
[18] Coatléven, J.. Joly, P., Operator factorization for multiple-scattering problems and an application to periodic media, Commun. Comput. Phys. 11 (2012) 303–318.CrossRefGoogle Scholar
[19] Deraemaeker, A., Babus˘ka, I., Bouillard, P., Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions, Int. J. Numer. Meth. Engng. 46 (1999) 471–499.3.0.CO;2-6>CrossRefGoogle Scholar
[20] Du, K., Li, B., Sun, W., A numerical study on the stability of a class of Helmholtz problems, J. Comput. Phys. 287 (2015) 46–59.CrossRefGoogle Scholar
[21] Enquist, B., Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977) 629–651.CrossRefGoogle Scholar
[22] Engquist, B., Ying, L., Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation, Commun. Pure Appl. Math. LXIV (2011) 697–735.CrossRefGoogle Scholar
[23] Feng, X., Li, Z., Qiao, Z., High order compact finite difference schemes for the Helmholtz equation with discontinuous coefficients, J. Comput. Math. 29 (2011) 324–340.CrossRefGoogle Scholar
[24] Feng, X., Wu, H., hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp. 80 (2011) 1997–2024.CrossRefGoogle Scholar
[25] Fu, Y., Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers, J. Comput. Math. 26 (2008) 98–111.Google Scholar
[26] Halpern, L., Trefethen, L., Wide-angle one-way wave equations, J. Acoust. Soc. Am. 84 (1988) 1397–1404.CrossRefGoogle ScholarPubMed
[27] Harari, I., Turkel, E., Accurate finite difference methods for time-harmonic wave propagation, J. Comput. Phys. 119 (1995) 252–270.CrossRefGoogle Scholar
[28] Henner, V., Belozaroza, T., Forinash, K., Mathematical methods in physics: Partial differential equations, fourier series, and special functions, A.K. Peters, Ltd., 2009.CrossRefGoogle Scholar
[29] Hsiao, G., Liu, F., Sun, J., Xu, L., A coupled BEM and FEM for the interior transmission problem in acoustics, J. Comput. Appl. Math. 235 (2011) 5213–5221.CrossRefGoogle Scholar
[30] Ihlenburg, F., Finite Element Analysis of Acoustic Scattering, Spring, NewYork, 1998.CrossRefGoogle Scholar
[31] Ihlenburg, F., Babus˘ka, I., Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, Int. J. Numer. Meth. Engng. 38 (1995) 3745–3774.CrossRefGoogle Scholar
[32] Ihlenburg, F., Babus˘ka, I, Finite element solution of the Helmholtz equation with high wave number part II: the h–p version of the FEM, SIAM J. Numer. Anal. 34 (1997) 315–358.CrossRefGoogle Scholar
[33] Ito, K., Qiao, Z., Toivanen, J., A domain decomposition solver for acoustic scattering by elastic objects in layered media. J. Comput. Phys. 227 (2008) 8685–8698.CrossRefGoogle Scholar
[34] Jo, C., Shin, C., Suh, J., An optimal 9-point, finite-difference, frequency-space 2-D scalar wave extrapolator, Geophysics 61 (1996) 529–537.CrossRefGoogle Scholar
[35] Kim, S., Zhang, H., Optimized schwarz method with complete radiation transmision comditions for the Helmholtz equation in waveguides, SIAM J. Numer. Anal. 53 (2015) 1537–1558.CrossRefGoogle Scholar
[36] Laird, A., Giles, M., Preconditioned iterative solution of the 2D Helmholtz equation, Technical report 02/12, Oxford University Computing Laboratory, Oxford, 2002.Google Scholar
[37] Lambe, L., Luczak, R., Nehrbass, J.. A new finite difference method for the Helmholtz equation using symbolic computation, Int. J. Comput. Engrg. Sci., 4 (2003) 121–144.Google Scholar
[38] Li, H., Ma, H., Sun, W., Legendre spectral Galerkin method for electromagnetic scattering from large cavities, SIAM J. Numer. Anal. 51 (2013) 353–376.CrossRefGoogle Scholar
[39] Ma, J., Zhu, J., Li, M., The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber, Engrg. Anal. Boundary Elem. 34 (2010) 1058–1063.CrossRefGoogle Scholar
[40] Melenk, J., Sauter, S., Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp. 79 (2010) 1871–1914.CrossRefGoogle Scholar
[41] Nabavia, M., Siddiqui, M., Dargahi, J., A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation, J. Sound Vibration 307 (2007) 972–982.CrossRefGoogle Scholar
[42] Operto, S., Virieux, J., Amestoy, P., L’Excellent, J., Giraud, L.. Ali, H., 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study, Geophysics 72 (2007) SM195-SM211.CrossRefGoogle Scholar
[43] Reps, B., Vanroose, W., Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer, Numer. Linear Algebra Appl. 19 (2012) 232–252.CrossRefGoogle Scholar
[44] Singer, I., Sixth-order accurate finite difference schemes for the Helmholtz equation, J. Comput. Acoust. 14 (2006) 339–351.CrossRefGoogle Scholar
[45] Singer, I., Turkel, E., High-order finite difference methods for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 163 (1998) 343–358.CrossRefGoogle Scholar
[46] Shen, J., Wang, L., Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal. 45 (2007) 1954–1978.CrossRefGoogle Scholar
[47] Shen, J., Wang, L., Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal. 43 (2005) 623–644.CrossRefGoogle Scholar
[48] Shin, C., Sohn, H., A frequency-space 2-D scalar wave exteapolator using extended 25-point finite-diffrence operator, Geophysics, 63 (1998) 289–296.CrossRefGoogle Scholar
[49] Strouboulis, T., Babus˘ka, I., Hidajat, R., The generalized finite element method for Helmholtz equation: Theory, computation, and open problems, Comput. Methods Appl. Mech. Engrg. 195 (2006) 4711–4731.CrossRefGoogle Scholar
[50] Sun, W., Wu, J., Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence, IMA J. Numer. Anal. 28 (2008) 580–597.CrossRefGoogle Scholar
[51] Sutmann, G., Compact finite difference schemes of sixth order for the Helmholtz equation, J. Comput. Appl. Math. 203 (2007) 15–31.CrossRefGoogle Scholar
[52] Tezaura, R., Kalashnikovab, I., Farhata, C., The discontinuous enrichment method for medium-frequency Helmholtz problems with a spatially variable wavenumber, Comput. Methods Appl. Mech. Engrg. 268 (2014) 126–140.CrossRefGoogle Scholar
[53] Turkel, E., Gordon, D., Gordon, R., Tsynkov, S., Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number, J. Comput. Phys. 232 (2013) 272–287.CrossRefGoogle Scholar
[54] Wang, K., Wong, Y.S., Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model. 11 (2014) 787–815.Google Scholar
[55] Wang, K., Wong, Y.S., Deng, J., Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys. 17 (2015) 779–807.CrossRefGoogle Scholar
[56] Wong, Y.S., Li, G., Exact finite difference schemes for solving Helmholtz equation at any wavenumber, Int. J. Numer. Anal. Model. Ser. B 2 (2011) 91–108.Google Scholar
[57] Wu, H., Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version, IMA J. Numer. Anal. 34 (2014) 1266–1288.CrossRefGoogle Scholar
[58] Wu, J., Wang, Y., Li, W., Sun, W., Toeplitz-type approximations to the Hadamard integral operator and their applications to electromagnetic cavity problems, Appl. Numer. Math. 58 (2008) 101–121.CrossRefGoogle Scholar
[59] Zhao, M., A fast high order iterative solver for the electromagnetic scattering by open cavities filled with the inhomogeneous media, Adv. Appl. Math. Mech. 5 (2013) 235–257.CrossRefGoogle Scholar
[60] Zhao, M., Qiao, Z., Tang, T., A fast high order method for electromagnetic scattering by large open cavities, J. Comput. Math. 29 (2011) 287–304.CrossRefGoogle Scholar
[61] Zhu, L., Burman, E., Wu, H., Continuous interior penalty finite element method for Helmholtz equation with high wave number: one dimensional analysis, arXiv:1211.1424v1 [math.NA].Google Scholar
[62] Zhu, L., Wu, H., Preasymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part II: hp-version, SIAM J. Numer. Anal. 51 (2013) 1828–1852.CrossRefGoogle Scholar