Fully Discrete High-Order DG Schemes for Waves: Dispersion and Observability

Yunzhang Li; Xiaoyang Wang; Enrique Zuazua

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arXiv · 2605.17464

Fully Discrete High-Order DG Schemes for Waves: Dispersion and Observability

Yunzhang Li·Xiaoyang Wang·Enrique Zuazua

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Abstract

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.

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BibTeX RIS

Yunzhang Li, Xiaoyang Wang, Enrique Zuazua. 2026-05-17. Fully Discrete High-Order DG Schemes for Waves: Dispersion and Observability. https://arxiv.org/abs/2605.17464

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