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

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...

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Bibliographic Details
Published in:Journal of scientific computing 2016-06, Vol.67 (3), p.883-908
Main Authors: Qian, Jianliang, Yuan, Lijun, Liu, Yuan, Luo, Songting, Burridge, Robert
Format: Article
Language:English
Subjects:
Accuracy
Algorithms
Amplitudes
Approximation
Asymptotic methods
Asymptotic properties
Asymptotic series
Computational Mathematics and Numerical Analysis
Eikonal equation
Helmholtz equations
Inhomogeneous media
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Neighborhoods
Point sources
Theoretical
Transport equations
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Summary: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.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-015-0111-7