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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.17464 |
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| _version_ | 1866913147125760000 |
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| author | Li, Yunzhang Wang, Xiaoyang Zuazua, Enrique |
| author_facet | Li, Yunzhang Wang, Xiaoyang Zuazua, Enrique |
| contents | This paper investigates the spectral structure, numerical dispersion, and observability of fully discrete approximations of the one-dimensional wave equation by $P^k$ (local) discontinuous Galerkin methods. Characterizing the coupled space-time numerical dispersion reveals a trapping mechanism that forces the group velocities of both physical and spurious modes to vanish at selected frequencies. We then establish an exponential blow-up of order $\exp(h^{-(1-\varepsilon)})$ for the observability constant under this trapping mechanism. To overcome this divergence for arbitrary $k$, we propose a spectral filtering strategy to restore uniform observability. Theoretical analysis and numerical experiments indicate that higher-order methods may facilitate this recovery by preserving a larger genuine physical frequency band, thereby reducing filtering cost and observation time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17464 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fully Discrete High-Order DG Schemes for Waves: Dispersion and Observability Li, Yunzhang Wang, Xiaoyang Zuazua, Enrique Optimization and Control This paper investigates the spectral structure, numerical dispersion, and observability of fully discrete approximations of the one-dimensional wave equation by $P^k$ (local) discontinuous Galerkin methods. Characterizing the coupled space-time numerical dispersion reveals a trapping mechanism that forces the group velocities of both physical and spurious modes to vanish at selected frequencies. We then establish an exponential blow-up of order $\exp(h^{-(1-\varepsilon)})$ for the observability constant under this trapping mechanism. To overcome this divergence for arbitrary $k$, we propose a spectral filtering strategy to restore uniform observability. Theoretical analysis and numerical experiments indicate that higher-order methods may facilitate this recovery by preserving a larger genuine physical frequency band, thereby reducing filtering cost and observation time. |
| title | Fully Discrete High-Order DG Schemes for Waves: Dispersion and Observability |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2605.17464 |