时域和频域波动方程的Hadamard积分器：通过蝴蝶算法的Eulerian表述
Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms
AI总结：
基于Kirchhoff-Huygens表示和Duhamel原理，提出利用蝴蝶算法压缩的Hadamard积分器，通过TFT和TFTF方法高效求解时频域波动方程，实现准最优复杂度下隐式构建焦散外波场并推进空间翻转波。
摘要：
从时域波动方程的Kirchhoff-Huygens表示和Duhamel原理出发，我们提出了一种新颖的蝴蝶压缩Hadamard积分器，用于求解非均匀介质中自伴波动方程的时域和频域问题。首先，我们将Hadamard ansatz的主项纳入Kirchhoff-Huygens表示，以构建短时有效的传播子。其次，利用时间上的Fourier变换，我们推导出频域中相应的Eulerian短时传播子；在此传播子的基础上，我们进一步开发了用于时域波动方程Cauchy问题的时频时（TFT）方法。第三，我们进一步针对相应的点源Helmholtz方程提出了时频时频（TFTF）方法，该方法可为给定频带内的所有角频率提供Helmholtz方程的Green函数。第四，为了高效实现TFT和TFTF方法，我们引入蝴蝶算法来压缩不同频率下的振荡积分核。因此，所提出的方法能够隐式地构建焦散之外的波场，并以准最优的计算复杂度和内存使用量在时间上自然地推进空间翻转波。此外，一旦构建完成，Hadamard积分器即可用于求解具有各种初始条件的时域波动方程以及具有不同点源的频域波动方程。二维波动方程的数值算例验证了所提方法的准确性和效率。
英文摘要：
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.
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