Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms

TL;DR

The paper develops butterfly-compressed Eulerian Hadamard integrators for high-frequency wave equations by unifying Hadamard's time-domain ansatz with Babich's frequency-domain representation through a TFT/TFTF framework. By formulating short-time HKH propagators in an Eulerian setting and applying butterfly-based low-rank compression (IDBF and HODBF), the authors achieve quasi-linear time and memory complexity while naturally handling caustics and overturning waves. The methods solve time-domain Cauchy problems and frequency-domain Helmholtz problems with a single precomputed kernel set, enabling efficient propagation for multiple initial conditions or point sources, and are validated through 2-D numerical experiments showing accuracy and scalability. The work offers a practical, scalable approach for high-frequency wave simulations in inhomogeneous media, with potential impact on seismic imaging, acoustics, and electromagnetics where caustics and complex wavefronts are prevalent.

Abstract

Starting from the Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using the Fourier transform in time, we derive the corresponding Eulerian short-time propagator in frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose the time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods.

Figures (16)

• Figure 1: Illustration of $\Omega_S^{\ell}$ and $\Omega_R^{\ell}$: The red square represents the source region $\Omega_S^{\ell}$, and the largest square divided into nine parts represents the corresponding receiver region $\Omega_R^{\ell}$
• Figure 2: Sinusoidal model. (a) The velocity; (b) Rays and wavefronts with source ${ \hbox{\boldmath $x$} }_0=[0.5,0.5]$. The thick blue lines represent equal-time wavefronts (traveltime contours) with the contour interval equal to $0.1$, and thin colored lines represent rays with different take-off angles
• Figure 3: Sinusoidal model. $u(t,{ \hbox{\boldmath $x$} })$, $t=0.5$: (a) TFT solution with $\beta=8$; (b) TFT solution with $\beta=16$; (c) TFT solution with $\beta=32$; (d) reference solution with $\beta=8$; (e) reference solution with $\beta=16$; (f) reference solution with $\beta=32$
• Figure 4: Sinusoidal model. Slices of time-domain wave fields $u(t,{ \hbox{\boldmath $x$} })$ when $t=0.5$: (a) a slice at $x=0.2$ with $\beta=8$; (b) a slice at $x=0.2$ with $\beta=16$; (c) a slice at $x=0.2$ with $\beta=24$; (d) a slice at $y=0.85$ with $\beta=8$; (e) a slice at $y=0.85$ with $\beta=16$; (f) a slice at $y=0.85$ with $\beta=24$
• Figure 5: Sinusoidal model. Time-domain point-source wave fields $u(t,{ \hbox{\boldmath $x$} })$. (a) $T=0.25$; (b) $T=0.5$; (c) $T=0.75$; (d) $T=1$; (e) $T=1.25$; (f) $T=1.5$