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.

翻译：从时域波动方程的基尔霍夫-惠更斯表示和杜阿梅尔原理出发，我们针对非均匀介质中的自伴波动方程，在时域和频域分别提出了新型的蝴蝶压缩Hadamard积分器。首先，将Hadamard假设的首项融入基尔霍夫-惠更斯表示，构建短时有效传播子。其次，利用时间傅里叶变换，推导出频域中对应的欧拉短时传播子；在此传播子基础上，进一步提出时-频-时（TFT）方法求解时域波动方程的柯西问题。第三，针对相应的点源亥姆霍兹方程，提出时-频-时-频（TFTF）方法，该方法可为给定频带内所有角频率提供亥姆霍兹方程的格林函数。第四，为高效实现TFT和TFTF方法，引入蝴蝶算法压缩不同频率下的振荡积分核。最终，所提方法能够隐式构建焦散面之外的波场，并以准最优的计算复杂度和内存占用自然地推进空间翻转波。此外，构建完成的Hadamard积分器可同时用于求解具有不同初始条件的时域波动方程和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精度与效率。