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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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Volume 26, Issue 2

Issues

Marching schemes for Cauchy wave propagation problems in laterally varying waveguides

Peng Li
  • Corresponding author
  • School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, P. R. China
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/ Keying Liu
  • School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011; and School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710061, P. R. China
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/ Weizhou Zhong
  • School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710061; and School of Business Administration, Huaqiao University, Quanzhou 362021, P. R. China
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Published Online: 2017-10-17 | DOI: https://doi.org/10.1515/jiip-2016-0044

Abstract

This paper intends to develop practical marching schemes for Cauchy problems of the Helmholtz equation in laterally varying waveguides. We arrive at a stable representation of the marching solutions in waveguides. Based on the representation, a second-order marching scheme is then constructed to eliminate the ill-conditioning and compute the wave propagation in waveguides with laterally variable mediums. In the end, extensive experiments are implemented to verify the efficiency and accuracy of the marching scheme in various waveguides, and we also point out the application scope of the scheme.

Keywords: Cauchy problem; propagating mode; marching method; waveguide; Helmholtz equation

MSC 2010: 65N12; 65N20; 65N35

References

  • [1]

    G. Beylkin and K. Sandberg, Wave propagation using bases for bandlimited functions, Wave Motion 41 (2005), no. 3, 263–291. CrossrefGoogle Scholar

  • [2]

    M. D. Collins and W. A. Kuperman, Inverse problems in ocean acoustics, Inverse Problems 10 (1994), no. 5, 1023–1040. CrossrefGoogle Scholar

  • [3]

    M. D. Collins and W. L. Siegmann, Parabolic Wave Equations with Applications, Springer, New York, 2001. Google Scholar

  • [4]

    T. Delillo, V. Isakov, N. Valdivia and L. Wang, The detection of the source of acoustical noise in two dimensions, SIAM J. Appl. Math. 61 (2001), no. 6, 2104–2121. CrossrefGoogle Scholar

  • [5]

    T. DeLillo, V. Isakov, N. Valdivia and L. Wang, The detection of surface vibrations from interior acoustical pressure, Inverse Problems 19 (2003), no. 3, 507–524. CrossrefGoogle Scholar

  • [6]

    L. Fishman, One-way wave propagation methods in direct and inverse scalar wave propagation modeling, Radio Sci. 28 (1993), 865–876. CrossrefGoogle Scholar

  • [7]

    F. B. Jensen, W. A. Kuperman, M. B. Porter and H. Schmidt, Computational Ocean Acoustics, AIP Ser. Modern Acoustics Signal Process., American Institute of Physics, New York, 1994. Google Scholar

  • [8]

    B. Jin and Y. Zheng, A meshless method for some inverse problems associated with the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 19–22, 2270–2288. CrossrefGoogle Scholar

  • [9]

    C. S. Kenney and A. J. Laub, The matrix sign function, IEEE Trans. Automat. Control 40 (1995), no. 8, 1330–1348. CrossrefGoogle Scholar

  • [10]

    C. H. Knightly and D. F. St. Mary, Stable marching schemes based on elliptic models of wave propagation, J. Acoustical Soc. Amer. 93 (1993), 1866–1872. CrossrefGoogle Scholar

  • [11]

    P. Li, Z. Chen and J. Zhu, An operator marching method for inverse problems in range-dependent waveguides, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 49–50, 4077–4091. CrossrefGoogle Scholar

  • [12]

    P. Li, K. Liu, W. Zuo and W. Zhong, Error analysis for the operator marching method applied to range dependent waveguides, J. Inverse Ill-Posed Probl. 24 (2016), no. 5, 625–636. Web of ScienceGoogle Scholar

  • [13]

    P. Li, W. Z. Zhong, G. S. Li and Z. H. Chen, A numerical local orthogonal transform method for stratified waveguides, J. Zhejiang Univ. Sci. (C) 11 (2010), no. 12, 998–1008. CrossrefWeb of ScienceGoogle Scholar

  • [14]

    Y. Y. Lu, One-way large range step methods for Helmholtz waveguides, J. Comput. Phys. 152 (1999), no. 1, 231–250. CrossrefGoogle Scholar

  • [15]

    Y. Y. Lu and J. R. McLaughlin, The Riccati method for the Helmholtz equation, J. Acoustical Soc. Amer. 100 (1996), 1432–1446. CrossrefGoogle Scholar

  • [16]

    Y. Y. Lu, J. Huang and J. R. McLauphlin, Local orthogonal transformation and one-way methods for acoustics waveguides, Wave Motion 34 (2001), 193–207. CrossrefGoogle Scholar

  • [17]

    Y. Y. Lu and J. Zhu, A local orthogonal transform for acoustic waveguides with an interval interface, J. Comput. Acoust. 12 (2004), no. 1, 37–53. CrossrefGoogle Scholar

  • [18]

    F. Natterer and F. Wübbeling, A propagation-backpropagation method for ultrasound tomography, Inverse Problems 11 (1995), no. 6, 1225–1232. CrossrefGoogle Scholar

  • [19]

    F. Natterer and F. Wübbeling, Marching schemes for inverse acoustic scattering problems, Numer. Math. 100 (2005), no. 4, 697–710. CrossrefGoogle Scholar

  • [20]

    B. N. Parlett, The Rayleigh quotient iteration and some generalizations for nonnormal matrices, Math. Comp. 28 (1974), 679–693. CrossrefGoogle Scholar

  • [21]

    K. Sandberg, Forward and Inverse Wave Propagation Using Bandlimited Functions and a Fast Reconstruction Algorithm for Electron Microscopy, ProQuest LLC, Ann Arbor, 2003. Google Scholar

  • [22]

    K. Sandberg and G. Beylkin, Full-wave-equation depth extrapolation for migrations, Geophys. 74 (2009), no. 6, 121–128. CrossrefGoogle Scholar

  • [23]

    K. Sandberg, G. Beylkin and A. Vassiliou, Full-wave-equation depth migration using multiple reflections, preprint (2010), http://www.geoenergycorp.com/publications/SEG2010_Two_Way_WEM.pdf.

  • [24]

    R. C. Song, J. X. Zhu and X. C. Zhang, Full-vectorial modal analysis for circular optical waveguides based on the multidomain Chebyshev pseudospectral method, J. Optical Soc. Amer. B 27 (2010), no. 9, 1722–1730. CrossrefGoogle Scholar

  • [25]

    M. Vögeler, Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method, Inverse Problems 19 (2003), no. 3, 739–753. CrossrefGoogle Scholar

  • [26]

    Z. Wang and S. F. Wu, Helmholtz equation-least-squares method for reconstructing the acoustic pressure field, J. Acoustical Soc. Amer. 102 (1997), no. 4, 2020–2032. CrossrefGoogle Scholar

  • [27]

    J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. Google Scholar

  • [28]

    S. F. Wu and J. Yu, Reconstructing interior acoustic pressure fields via Helmholtz equation least-squares method, J. Acoustical Soc. Amer. 104 (1998), no. 4, 2054–2060. CrossrefGoogle Scholar

  • [29]

    X. Zhang, Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm–Liouville problems, Appl. Math. Comput. 217 (2010), no. 5, 2266–2276. Web of ScienceGoogle Scholar

  • [30]

    J. Zhu and P. Li, Local orthogonal transform for a class of acoustic waveguides, Progr. Natur. Sci. (English Ed.) 17 (2007), no. 10, 1136–1146. Google Scholar

  • [31]

    J. Zhu and Y. Lu, Large range step method for acoustic waveguide with two layer media, Progr. Natur. Sci. (English Ed.) 12 (2002), no. 11, 820–825. Google Scholar

  • [32]

    J. Zhu and Y. Y. Lu, Validity of one-way models in the weak range dependence limit, J. Comput. Acoust. 12 (2004), no. 1, 55–66. CrossrefGoogle Scholar

  • [33]

    J. X. Zhu and P. Li, The mathematical treatment of wave propagation in the acoustical waveguides with n curved interfaces, J. Zhejiang Univ. Sci. (A) 9 (2008), no. 10, 1463–1472. CrossrefGoogle Scholar

  • [34]

    J. X. Zhu and R. C. Song, Fast and stable computation of optical propagation in micro-waveguides with loss, Mircroelectronics Reliab. 49 (2009), no. 12, 1529–1536. CrossrefGoogle Scholar

  • [35]

    J. X. Zhu and Q. X. Zhou, Eigenvalue computation in slab waveguides with some perfectly matched layer, Proc. SPIE 5279 (2004), 172–180. CrossrefGoogle Scholar

About the article

Received: 2015-09-04

Revised: 2017-09-10

Accepted: 2017-09-16

Published Online: 2017-10-17

Published in Print: 2018-04-01


This research was funded in part by the China Scholarship Council No. 201408410160, the Science and Technology Research Project of the Education Department of Henan Province 12A110014.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 26, Issue 2, Pages 259–276, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2016-0044.

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