Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T01:05:19.011Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fast High Order Iterative Solver for the Electromagnetic Scattering by Open Cavities Filled with the Inhomogeneous Media

Published online by Cambridge University Press:  03 June 2015

Meiling Zhao
Meiling Zhao*
Affiliation:
School of Mathematics & Physics, North China Electric Power University, Baoding, 071003, China School of Mathematics and System Sciences, Beijing University of Aeronautics & Astronautics, Beijing, 100083, China
*
Corresponding author. Email: [email protected]
Get access

Abstract

The scattering of the open cavity filled with the inhomogeneous media is studied. The problem is discretized with a fourth order finite difference scheme and the immersed interface method, resulting in a linear system of equations with the high order accurate solutions in the whole computational domain. To solve the system of equations, we design an efficient iterative solver, which is based on the fast Fourier transformation, and provides an ideal preconditioner for Krylov subspace method. Numerical experiments demonstrate the capability of the proposed fast high order iterative solver.

Keywords

65N0678M20Helmholtz equationcompact finite difference schemediscontinuous wave numbersimmerse interface methodfast iterative solver
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 2 , April 2013 , pp. 235 - 257
Copyright
Copyright © Global-Science Press 2013

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]Ihlenburg, F., Finite element analysis of acoustic scattering, Applied Mathematical Sciences, Springer-Verlag, New York, 1998.Google Scholar
[2]Liu, J. and Jin, J. M., A special high-order finite element method for scattering by deep cavity, IEEE Tran. Antennas Propag., 48 (2000), pp. 694–703.Google Scholar
[3]Ihlenburg, F. and Babuska, I., Finite element solution of the Helmholtz equation with high wave number Part I: the h-version of FEM, Comput. Math. Appl., 30 (1995), pp. 9–37.Google Scholar
[4]Ihlenburg, F. and Babuska, 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), pp. 315–358.Google Scholar
[5]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), pp. 8685–8698.CrossRefGoogle Scholar
[6]Fang, Q., Nicholls, D. P. and Shen, J., A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering, J. Comput. Phys., 224 (2007), pp. 1145–1169.CrossRefGoogle Scholar
[7]Fu, Y., Compact fourth-order finite difference schemes for helmholtz equation with high wave numbers. J. Comput. Math., 26(2008), pp.98–111.Google Scholar
[8]Gustafsson, B. and Mossberg, E., Time compact high order difference methods for wave propagation, SIAM J. Sci. Comput., 26 (2004), pp. 259–271.CrossRefGoogle Scholar
[9]Baruch, G., Fibich, G., Tsynkov, S. and Turkel, E., Fourth order schemes for time-harmonic wave equations with discontinuous coefficients, Commun. Comput. Phys., 5 (2009), pp. 442–455.Google Scholar
[10]Baruch, G., Fibich, G. and Tsynkov, S., High-order numerical method for the nonlinear Helmholtz equation with material discontinuities, J. Comput. Phys., 227 (2007), pp. 820–850.Google Scholar
[11]Chen, Q., Monk, P., Wang, X. and Weile, D., Analysis of convolution quadrature applied to the time-domain electric field integral equation, Commun. Comput. Phys., 11 (2012), pp. 383–399.CrossRefGoogle Scholar
[12]Ito, K. and Qiao, Z., A high order compact MAC finite difference scheme for the Stokes equations: augmented variable approach, J. Comput. Phys., 227 (2008), pp. 8177–8190.CrossRefGoogle Scholar
[13]Ito, K. and Qiao, Z., A high order finite difference scheme for the Stokes equations, AMS Contem. Math., 466 (2008), pp. 35–51.CrossRefGoogle Scholar
[14]Jin, J., Liu, J., Lou, Z. and Liang, S., A fully high-order finite-element simulation of scattering by deep cavities, IEEE Trans. Antennas Propag., 51 (2003), pp. 2420–2429.Google Scholar
[15]Callihan, R. S. and Wood, A. W., A modified Helmholtz equation with impedance boundary conditions, Adv. Appl. Math. Mech., 4 (2012), pp. 703–718.Google Scholar
[16]Leveque, R. J. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 1019–1044.CrossRefGoogle Scholar
[17]Li, Z. and Ito, K., The immersed interface method-numerical solutions of pdes involving interfaces and irregular domains, SIAM Frontier Series in Applied Mathematics, FR33, 2006.Google Scholar
[18]Li, Z. and Lai, M.-C., New finite difference methods based on IIM for inextensible interfaces in incompressible flows, East Asian J. Appl. Math., 1 (2011), pp. 155–171.CrossRefGoogle ScholarPubMed
[19]Zhao, M., Qiao, Z. and Tang, T., A fast high order method for electromagnetic scattering by large open cavities, J. Comput. Math., 29 (2011), pp. 287–304.Google Scholar
[20]Li, C. and Qiao, Z., A fast preconditioned iterative algorithm for the electromagnetic scattering from a large cavity, J. Sci. Comput., 53 (2012), pp. 435–450.Google Scholar
[21]Bao, G., Gao, J. and Li, P., Analysis of direct and inverse cavity scattering problems, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 335–358.Google Scholar
[22]Plessix, R. E. and Mulder, W. A., Separation of variables as a preconditioner for an iterative Helmholtz solver, Appl. Numer. Math., 44 (2003), pp. 385–400.CrossRefGoogle Scholar
[23]Larid, A. L. and Giles, M. B., Preconditioned iterative solution of the 2D Helmholtz equation, Report NA-02/12, Oxford University Computing Laboratory, 2002.Google Scholar
[24]Erlangga, Y. A., Vuik, C. and Oosterlee, C. W., On a class of preconditioners for solving the Helmholtz equation, Appl. Numer. Math., 50 (2004), pp. 409–425.Google Scholar
[25]Saad, Y., ILUT: a dual threshold incomplete LU factorization, Numer. Linear Algebra Appl., 4 (1994), pp. 387–402.Google Scholar
[26]Bialecki, B., Fairweather, G. and Karageorghis, A., Matrix decomposition algorithms for elliptic boundary value problems: a survey, Numer. Algor., 56 (2011), pp. 253–295.Google Scholar
[27]Erlangga, Y. A., Advances in iterative methods and preconditioners for the Helmholtz equation, Arch. Comput. Methods Eng., 15 (2008), pp. 37–66.Google Scholar
[28]Lee, J., Zhang, J. and Lu, C. C., Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems, J. Comput. Phys., 185 (2003), pp. 158–175.Google Scholar
[29]Sheng, Q. and Sun, H.-W., Asymptotic stability of an eikonal transformation based ADI method for the paraxial Helmholtz equation at high wave numbers, Commun. Comput. Phys., 12 (2012), pp. 1275–1292.Google Scholar
[30]Kuffuor, D. O. and Saad, Y., Preconditioning Helmholtz linear systems, Appl. Numer. Math., 60 (2010), pp. 420–431.Google Scholar
[31]Ito, K. and Toivanen, J., A fast iterative solver for scattering by elasitc objects in layered media, Appl. Numer. Math., 57 (2007), pp. 811–820.Google Scholar
[32]Feng, X., Qiao, Z. and Li, Z., High order compact finite difference schemes for Helmholtz equation with discontinuous coefficient. J. Comput. Math., 29 (2011), pp.324–340.Google Scholar
[33]Wu, J., Wang, Y., Li, W. and Sun, W., Toeplitz-type aooroximations to the Hadamard integral operators and their applications in electromagnetic cavity problems, Appl. Numer. Math., 58 (2008), pp. 101–121.Google Scholar
[34]Smith, B. F., Bjorstad, P. E. and Groop, W. D., Domain Decomposition, Cambridge University Press, Cambridge, 1996.Google Scholar
[35]Ito, K. and Toivanen, J., Preconditioned iterative methods on sparse subspaces, Appl. Math. Lett., 19 (2006), pp. 1191–1197.CrossRefGoogle Scholar
[36]Chan, R. H. and Ng, M. K., Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427–482.Google Scholar
[37]Bao, G. and Sun, W., A fast algorithm for the electromagnetic scattering from a large cavity, SIAM J. Sci. Comput., 27 (2005), pp. 553–574.Google Scholar
[38]Wang, Y., Du, K. and Sun, W., A second-order method for the electromagnetic scattering from a large cavity, Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 357–382.Google Scholar
[39]Axelsson, O., Iterative Solution Meshods, Cambridge University Press, New York, 1994.CrossRefGoogle Scholar