Asymptotic Solutions for High Frequency Helmholtz Equations in Anisotropic Media with Hankel Functions

  • Matthew Jacobs
  • Songting LuoEmail author


We present asymptotic methods for solving high frequency Helmholtz equations in anisotropic media. The methods are motivated by Babich’s expansion that uses Hankel functions of the first kind to approximate the solution of high frequency Helmholtz equation in isotropic media. Within Babich’s expansion, we can derive the anisotropic eikonal equation and a recurrent system of transport equations to determine the phase and amplitude terms of the wave, respectively. In order to reconstruct the wave with the phase and amplitude terms for any high frequencies, they must be computed with high-order accuracy, for which a high-order factorization approach based on power series expansions at the primary source is applied first to resolve the source singularities, after that high-order schemes can be implemented efficiently. Rigorous formulations are derived, and numerical examples are presented to demonstrate the methods.


Anisotropic Helmholtz equation Asymptotic approximation Babich’s expansion Anisotropic eikonal equation Source singularity High-order factorization High-order scheme 

Mathematics Subject Classification

65N06 41A60 



Funding was provided by NSF Division of Mathematical Sciences (1418908, 1719907).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

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