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 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 geometricaloptics 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 B Jianliang Qian [email protected] Lijun Yuan [email protected] Yuan Liu [email protected] Songting Luo [email protected] Robert Burridge [email protected] 1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 2 College of Math and Statistics, Chongqing Technology and Business University, Chongqing 400067, People’s Republic of China 3 Department of Mathematics, Iowa State University, Ames, IA 50011, USA 4 Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA