采用MAC格式离散的各向同性弹性Helmholtz方程的渐近频散校正
Asymptotic dispersion correction for the isotropic elastic Helmholtz equation discretized with a MAC scheme
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中文总结 AI 辅助
针对弹性Helmholtz方程MAC离散的高频数值频散问题，提出渐近最优频散校正方法，可降低相对误差并改善多重网格求解器收敛性，为弹性问题代数结构提供新见解。
中文摘要 AI 辅助
弹性介质中时谐波传播的数值模拟在地球物理、无损检测等应用中具有重要作用。由于数值频散与污染效应，高频下弹性Helmholtz方程的精确离散颇具挑战性。本研究针对各向同性弹性Helmholtz方程的Marker-And-Cell（MAC）离散格式，提出一种渐近频散校正方法。我们刻画了该格式的离散频散关系，确定了二维与三维空间下频散误差的主导阶项；基于此分析，推导了在网格尺寸趋于零的极限下渐近最优的校正项，该方法在保留底层离散结构的同时，提升了离散与连续波传播特性的一致性。我们还建立了频散关系因式分解与grad-div算子符号结构的关联，为弹性问题的代数结构提供了进一步见解。数值实验最终表明，该方法可大幅降低相对误差，验证了校正的有效性；我们还提供数值证据，证明经校正的离散格式可改善多重网格求解器的收敛行为。
英文摘要
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.
