Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929373882351616 |
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| author | Wei, Yuxiao Cheng, Jin Leung, Shingyu Burridge, Robert Qian, Jianliang |
| author_facet | Wei, Yuxiao Cheng, Jin Leung, Shingyu Burridge, Robert Qian, Jianliang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01423 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms Wei, Yuxiao Cheng, Jin Leung, Shingyu Burridge, Robert Qian, Jianliang Numerical Analysis Mathematical Physics 65M80, 65Y20 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. |
| title | Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms |
| topic | Numerical Analysis Mathematical Physics 65M80, 65Y20 |
| url | https://arxiv.org/abs/2401.01423 |