Babich's expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies

Abstract

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation can be viewed as an evolution equation in one of the spatial directions. With such applications in mind, starting from Babich's expansion, we develop a new high-order asymptotic method, which we dub the fast Huygens sweeping method, for solving point-source Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of this method is that we develop a new Eulerian approach to compute the asymptotics, i.e. the traveltime function and amplitude coefficients that arise in Babich's expansion, yielding a locally valid solution, which is accurate close enough to the source. The second novelty is that we utilize the Huygens-Kirchhoff integral to integrate many locally valid wavefields to construct globally valid wavefields. This automatically treats caustics and yields uniformly accurate solutions both near the source and remote from it. The third novelty is that the butterfly algorithm is adapted to accelerate the Huygens-Kirchhoff summation, achieving nearly optimal complexity O(N log N ), where N is the number of mesh points; the complexity prefactor depends on the desired accuracy and is independent of the frequency. To reduce the storage of the resulting tables of asymptotics in Babich's expansion, we use the multivariable Chebyshev series expansion to compress each table $ Qian is supported by NSF grants 1222368 and 1522249.

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

“…This has been justified in [62] for oscillatory initial value problems of hyperbolic equations and further made rigorous in the theory of Fourier integral operators [48]. In practice, the one-term asymptotic expansion (7), namely the so-called geometric optics term, usually yields sufficiently accurate asymptotic solutions [1,2,59,65,66,85,86].…”

“…Under the assumption that the medium is smooth and no caustic occurs, one may solve the transport equations to estimate the coefficients {A l } in different formulations [1,2,65]. Since the geometric optics term is oscillatory when ω = 0, it should be understood in the L 2 sense rather than the L ∞ sense.…”

Extending Babich's Ansatz for Point-Source Maxwell's Equations Using Hadamard's Method

“…As explained and illustrated in [16,17,22,23], given a primary source point, we a priori partition the computational domain into several layers as sketched in Figure 1, where r 0 denotes the primary source, S m is the mth secondary source plane z = z m so that Ω m+1 = {(x, y, z) T |z m < z < z m+1 }, the wavefield associated with the primary source r 0 has no caustics in the region Ω 0 but may develop caustics beyond Ω 0 , and, for all m, the wavefields associated with the secondary sources situated on S m have no caustics in Ω m+1 but may develop caustics beyond Ω m+1 .…”

“…It is worth mentioning that his method of finding such an ansatz is closely bound up with Hadamard's method of forming the fundamental solution of the Cauchy problem for the time-domain point-source (TDPS) Helmholtz wave equation; details were given in [9] and then were outlined by Courant and Hilbert [5]. Recently, Babich's ansatz has been applied to numerically solve the FDPS Helmholtz equation in [16,25]. Nevertheless, Babich's ansatz cannot be trivially extended to the FDPS Maxwell's equations (1.1).…”

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

“…On the other hand, the phase φ and amplitude coefficients v 0 , v 1 are numerically computed by Algorithm 2. According to Theorem 5.1 in [61] and Remark 3 in [46], the p-th order LxF-WENO scheme combines with q-th order factorization for equations (15) and (16) yield min(p, q)-th order accuracy for smooth φ and v's. Thus, we have…”