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 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.