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-惠更斯表示法和杜阿梅尔原理出发，针对非均匀介质中的自伴波动方程，我们提出了新型蝴蝶压缩的Hadamard积分器，适用于时域和频域两种场景。首先，将Hadamard假设的主导项融入基尔霍夫-惠更斯表示法，构建了短时有效传播子；其次，利用时间傅里叶变换，推导了频域中相应的欧拉短时传播子，并在此基础上进一步提出时-频-时方法，求解时域波动方程的柯西问题；第三，针对对应的点源亥姆霍兹方程，提出时-频-时-频方法，能够提供给定频带内所有角频率的格林函数；第四，为高效实现时-频-时和时-频-时-频方法，引入蝴蝶算法对不同频率处的振荡积分核进行压缩。由此，所提方法能以准最优计算复杂度和内存占用，隐式构建焦散面以外的波场，并自然推进时域中空间翻转波形。一旦构建完成，该Hadamard积分器可同时用于求解不同初始条件下的时域波动方程和不同点源的频域波动方程。二维波动方程数值算例验证了所提方法的准确性与高效性。