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Efficient and Accurate Numerical Solutions for Helmholtz Equation in Polar and Spherical Coordinates

Published online by Cambridge University Press:  24 March 2015

Kun Wang ,
Yau Shu Wong  and
Jian Deng
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
Yau Shu Wong
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
Jian Deng
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada
*
*Corresponding author. Email addresses: [email protected] (K. Wang), [email protected] (Y. S. Wong), [email protected] (J. Deng)
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Abstract

This paper presents new finite difference schemes for solving the Helmholtz equation in the polar and spherical coordinates. The most important result presented in this study is that the developed difference schemes are pollution free, and their convergence orders are independent of the wave number k. Let h denote the step size, it will be demonstrated that when solving the Helmholtz equation at large wave numbers and considering kh is fixed, the errors of the proposed new schemes decrease as h decreases even when k is increasing and kh > 1.

Keywords

65N0665N1565N22Helmholtz equationhigh wave numberpolar coordinatespherical coordinatepollution freehigh order difference scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 779 - 807
Copyright
Copyright © Global-Science Press 2015 

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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]Babuška, I. and Banerjee, U., Stable generalized finite element method (SGFEM), Comput. Methods Appl. Mech. Engrg., 201–204 (2012), 91–111.Google Scholar
[3]Zarmi, A. and Turkel, E., A general approach for high order absorbing boundary conditions for the Helmholtz equation, J. Comput. Phys., 242 (2013), 387–404.CrossRefGoogle Scholar
[4]Tezaura, R., Kalashnikovab, I. and 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.Google Scholar
[5]Babuška, I., Ihlenburg, F.Paik, E. and 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.Google Scholar
[6]Babuška, I. and 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
[7]Britt, S., Tsynkov, S. and Turkel, E., A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates, J. Sci. Comput., 45 (2010), 26–47.Google Scholar
[8]Turkel, E., Gordon, D.Gordon, R. and Tsynkov, S., Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number, J. Comput. Phys., 232 (2013), 272–287.Google Scholar
[9]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.Google Scholar
[10]Ito, K., Qiao, Z. and Toivanen, J., A domain decomposition solver for acoustic scattering by elastic objects in layered media, J. Comput. Phys., 227 (2008), 8685–8698.Google Scholar
[11]Callihan, R. and Wood, A., A modified Helmholtz equation with impedance boundary conditions, Adv. Appl. Math. Mech., 4 (2012), 703–718.Google Scholar
[12]Chen, H., Lu, P. and Xu, X., A hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number, SIAM J. Numer. Anal., 51 (2013), 2166–2188.CrossRefGoogle Scholar
[13]Jo, C., Shin, C. and Suh, J., An optimal 9-point, finite-difference, frequency-space 2-D scalar wave extrapolator, Geophysics, 61 (1996), 529–537.Google Scholar
[14]Operto, S., Virieux, J., Amestoy, P., L’Excellent, J. and 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.Google Scholar
[15]Bao, G., Wei, G.W. and Zhao, S., Numerical solution of the helmholtz equation with high wavenumbers, Int. J. Numer. Meth. Engng., 59 (2004), 389–408.CrossRefGoogle Scholar
[16]Deraemaeker, A., Babuška, I. and 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
[17]Feng, X. and Wu, H., hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), 1997–2024.Google Scholar
[18]Fu, Y., Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers, J. Comput. Math., 26 (2008), 98–111.Google Scholar
[19]Hsiao, G., Liu, F., Sun, J. and Xu, L., A coupled BEM and FEM for the interior transmission problem in acoustics, J. Comput. Appl. Math., 235 (2011), 5213–5221.Google Scholar
[20]Ma, J., Zhu, J. and 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.Google Scholar
[21]Ihlenburg, F., Finite Element Analysis of Acoustic Scattering, Spring, NewYork, 1998.Google Scholar
[22]Morse, P. and Feshbach, H., Methods of Theoretical Physics, Part I: Chapters 1 to 8, Mcgraw-Hill Book Company, New York, 1953.Google Scholar
[23]Jones, D., Acoustic and Electromagnetic Waves, Clarendon Press, Oxford, 1986.Google Scholar
[24]Ihlenburg, F. and Babuška, 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
[25]Ihlenburg, F. and Babuška, I., Finite element solution of the Helmholtz equation with high wave number part I: The h-version of the FEM, Comput. Math. Appl., 30 (1995), 9–37.Google Scholar
[26]Lambe, L., Luczak, R. and 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
[27]Shen, J. and 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
[28]Shen, J. and Wang, L., Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), 623–644.Google Scholar
[29]Singer, I., Sixth-order accurate finite difference schemes for the Helmholtz equation, J. Comput. Acoust., 14 (2006), 339–351.Google Scholar
[30]Strouboulis, T., Babuška, I. and 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
[31]Singer, I. and Turkel, E., High-order finite difference methods for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 163 (1998), 343–358.CrossRefGoogle Scholar
[32]Chen, Z., Cheng, D. and Wu, T., A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation, J. Comput. Phys., 231 (2012), 8152–8175.Google Scholar
[33]Wong, Y.S. and 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
[34]Wang, K. and Wong, Y.S., Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model., 11 (2014), 787–815.Google Scholar
[35]Chen, Z., Cheng, D., Feng, W. and 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
[36]Weisstein, E.W., Helmholtz Differential Equation–Polar Coordinates, From Math World–A Wolfram Web Resource.Google Scholar
[37]Zhai, S., Weng, Z. and Feng, X., High-order compact operator splitting method for three-dimensional fractional equation with subdiffusion, submitted.Google Scholar
[38]Esterhazy, S. and Melenk, J., On Stability of Discretizations of the Helmholtz Equation, 285–324. In Graham, I.G.et al. (eds.), Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engineering 83, Springer-Verlag, Berlin Heidelberg, 2012.CrossRefGoogle Scholar