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Babich’s Expansion and High-Order Eulerian Asymptotics for Point-Source Helmholtz Equations

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Abstract

The usual geometrical-optics expansion of the solution for the Helmholtz equation of a point source in an inhomogeneous medium yields two equations: an eikonal equation for the traveltime function, and a transport equation for the amplitude function. However, two difficulties arise immediately: one is how to initialize the amplitude at the point source as the wavefield is singular there; the other is that in even-dimension spaces the usual geometrical-optics expansion does not yield a uniform asymptotic approximation close to the source. Babich (USSR Comput Math Math Phys 5(5):247–251, 1965) developed a Hankel-based asymptotic expansion which can overcome these two difficulties with ease. Starting from Babich’s expansion, we develop high-order Eulerian asymptotics for Helmholtz equations in inhomogeneous media. Both the eikonal and transport equations are solved by high-order Lax–Friedrichs weighted non-oscillatory (WENO) schemes. We also prove that fifth-order Lax–Friedrichs WENO schemes for eikonal equations are convergent when the eikonal is smooth. Numerical examples demonstrate that new Eulerian high-order asymptotic methods are uniformly accurate in the neighborhood of the source and away from it.

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Acknowledgments

Qian is supported by NSF Grants DMS-1222368 and DMS-1522249. Yuan is partially supported by the National Natural Science Foundation of China (Project No. 11201508).

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian & Yuan Liu

  2. College of Math and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People’s Republic of China

    Lijun Yuan

  3. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Songting Luo

  4. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA

    Robert Burridge

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  1. Jianliang Qian
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  2. Lijun Yuan
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  3. Yuan Liu
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Correspondence to Jianliang Qian.

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Qian, J., Yuan, L., Liu, Y. et al. Babich’s Expansion and High-Order Eulerian Asymptotics for Point-Source Helmholtz Equations. J Sci Comput 67, 883–908 (2016). https://doi.org/10.1007/s10915-015-0111-7

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  • DOI: https://doi.org/10.1007/s10915-015-0111-7

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