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arXiv · 2608.17909

Asymptotic dispersion correction for the isotropic elastic Helmholtz equation discretized with a MAC scheme

Pierre-Henri Cocquet·Antoine Tonnoir·Rachel Yovel
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Abstract

The numerical simulation of time-harmonic wave propagation in elastic media plays an important role in applications such as geophysics and non-destructive testing. Accurate discretization of the elastic Helmholtz equation at high frequencies is challenging due to numerical dispersion and pollution effects. In this work, we develop an asymptotic dispersion correction for a Marker-And-Cell (MAC) discretization of the isotropic elastic Helmholtz equation. We characterize the discrete dispersion relation of the scheme and determine the leading-order term in the dispersion error in both two and three spatial dimensions. Based on this analysis, we derive a correction that is asymptotically optimal in the limit of vanishing mesh size. The proposed approach improves the agreement between the discrete and continuous wave propagation properties while preserving the structure of the underlying discretization. We also establish a connection between the factorization of the dispersion relation and the structure of the grad-div operator symbol, providing additional insight into the algebraic structure of the elastic problem. Numerical experiments finally demonstrate a substantial reduction of relative errors and confirm the effectiveness of the proposed correction. We further provide numerical evidence that the corrected discretization improves the convergence behavior of multigrid solvers.

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Pierre-Henri Cocquet, Antoine Tonnoir, Rachel Yovel. 2026-08-18. Asymptotic dispersion correction for the isotropic elastic Helmholtz equation discretized with a MAC scheme. https://arxiv.org/abs/2608.17909

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