Error analysis for the operator marching method applied to range dependent waveguides
-
Peng Li
Abstract
The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To explain this phenomenon, this study intends to present an error estimate for the OMM in range dependent waveguides. We utilize “approximation level” to measure truncation error for a marching method in various range step sizes. Then, the error estimate is developed to analyze the performances of the OMM. Through an error analysis, we verify the following features of the OMM: (i) it is valid to apply the OMM in slowly varying waveguides with very large range step sizes; (ii) the OMM may blow up suddenly when the range dependence is strong and the step size is extremely small in the same time. We also develop a three-number set to describe the stability and accuracy level of a general marching method for computing wave propagation in a waveguide. In the end, extensive numerical experiments are implemented to verify the correctness of the error analysis.
Funding source: Xi’an Jiaotong University
Award Identifier / Grant number: NCET-08-0450
Award Identifier / Grant number: 985 II
Funding statement: Supported by NCET-08-0450 and 985 II of Xi’an Jiaotong University.
References
[1] Collins M. D. and Kuperman W. A., Inverse problems in ocean acoustics, Inverse Problems 10 (1994), 1023–1040. 10.1088/0266-5611/10/5/003Search in Google Scholar
[2] Collins M. D. and Siegmann W. L., Parabolic Wave Equations with Applications, Springer, New York, 2001. Search in Google Scholar
[3] Delillo T., Isakov V., Valdivia N. and Wang L., The detection of the source of acoustical noise in two dimensions, SIAM J. Appl. Math. 61 (2001), 2104–2121. 10.1137/S0036139900367152Search in Google Scholar
[4] Delillo T., Isakov V., Valdivia N. and Wang L., The detection of surface vibrations from interior acoustical pressure, Inverse Problems 19 (2003), 507–524. 10.1088/0266-5611/19/3/302Search in Google Scholar
[5] Jensen F. B., Kuperman W. A., Porter M. B. and Schmidt H., Computational Ocean Acoustics, AIP, Melville, 1994. 10.1063/1.2808704Search in Google Scholar
[6] Knightly C. H. and Mary D. F. St., Stable marching schemes based on elliptic models of wave propagation, J. Acoust. Soc. A 93 (1993), 1866–1872. 10.1121/1.406701Search in Google Scholar
[7] Li P., Chen Z. H. and Zhu J. X., An operator marching method for inverse problems in range dependent waveguides, Comput. Methods Appl. Mech. Engrg. 197 (2008), 4077–4091. 10.1016/j.cma.2008.04.001Search in Google Scholar
[8] Li P. and Zhong W., The inverse fundamental operator marching method for Cauchy problems in range-dependent stratified waveguides, Math. Probl. Eng. 2011 (2011), Article ID 259479. 10.1155/2011/259479Search in Google Scholar
[9] Li P., Zhong W., Li G. S. and Chen Z. H., A numerical local orthogonal transform method for stratified waveguides, J. Zhejiang Univ. Sci. (C) 11 (2010), no. 12, 998–1008. 10.1631/jzus.C0910732Search in Google Scholar
[10] Lu Y. Y., One-way large range step methods for Helmholtz waveguides, J. Comput. Phys. 152 (1999), 231–250. 10.1006/jcph.1999.6243Search in Google Scholar
[11] Lu Y. Y., Huang J. and McLauphlin J. R., Local orthogonal transformation and one-way methods for acoustics waveguides, Wave Motion 34 (2001), 193–207. 10.1016/S0165-2125(00)00083-4Search in Google Scholar
[12] Lu Y. Y. and McLaughlin J. R., The Riccati method for the Helmholtz equation, J. Acoust. Soc. Amer. 100 (1996), 1432–1446. 10.1121/1.415990Search in Google Scholar
[13] Lu Y. Y. and Zhu J. X., A local orthogonal transform for acoustic waveguides with an internal interface, J. Comput. Phys. 12 (2004), 37–53. 10.1142/S0218396X04002183Search in Google Scholar
[14] Natterer F. and Wübbeling F., A propagation-backpropagation method for ultrasound tomography, Inverse Problems 11 (1995), 1225–1232. 10.1088/0266-5611/11/6/007Search in Google Scholar
[15] Natterer F. and Wübbeling F., Marching schemes for inverse acoustic scattering problems, Numer. Math. 100 (2005), 697–710. 10.1007/s00211-004-0580-3Search in Google Scholar
[16] Sandberg K. and Beylkin G., Full wave-equation depth extrapolation for migration, Geophys. 74 (2009), no. 6, 121–128. 10.1190/1.3202535Search in Google Scholar
[17] Zhu J. X. and Li P., Local orthogonal transform for a class of acoustic waveguide, Progr. Natural Sci. 17 (2007), 18–28. Search in Google Scholar
[18] Zhu J. X. and Li P., 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. 10.1631/jzus.A0720064Search in Google Scholar
[19] Zhu J. X. and Lu Y. Y., Large range step method for acoustic waveguide with two layer media, Progr. Natural Sci. 12 (2002), 820–825. Search in Google Scholar
[20] Zhu J. X. and Lu Y. Y., Validity of one-way models in the weak range dependence limit, J. Comput. Phys. 12 (2004), 55–66. 10.1142/S0218396X0400216XSearch in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- A finite element method for the inverse problem of boundary data recovery in an oxygen balance model
- On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation
- Application of the factorization method to retrieve a crack from near field data
- Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions
- The variational formulation of an inverse problem for multidimensional nonlinear time-dependent Schrödinger equation
- Integral identity for a class of ill-posed problems generated by a parabolic equation
- A meshless method to the solution of an ill-posed problem
- About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication
- Error analysis for the operator marching method applied to range dependent waveguides
Articles in the same Issue
- Frontmatter
- A finite element method for the inverse problem of boundary data recovery in an oxygen balance model
- On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation
- Application of the factorization method to retrieve a crack from near field data
- Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions
- The variational formulation of an inverse problem for multidimensional nonlinear time-dependent Schrödinger equation
- Integral identity for a class of ill-posed problems generated by a parabolic equation
- A meshless method to the solution of an ill-posed problem
- About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication
- Error analysis for the operator marching method applied to range dependent waveguides
