论文标题
纠正深水模型的表面重力波
Rectification of a deep water model for surface gravity waves
论文作者
论文摘要
在这项工作中，我们讨论了一个近似模型，以传播深度无关的水波，特别是通过在Zakharov/Craig-Sulem公式下仅将二次非线性保留在水波系统中获得的模型。我们认为，与该系统相关的初步值问题在有限的规律性空间中很可能是不适合的，并且它解释了文献中报道的数值模拟中高波动模式的虚假扩增的观察。 Ambrose，Bona和Nicholls [4]已经提出了这一假设，但我们确定了不同的不稳定性机制。在此分析的基础上，我们表明该系统可以“纠正”。确实，通过引入适当的正规化操作员，我们可以恢复适当的性能，而无需牺牲其他理想的特征，例如规范的哈密顿结构，立方准确性作为渐近模型以及有效的数值集成。这为在高阶光谱方法中应用过滤器的常见实践提供了第一个严格的理由，以用于表面重力波的数值近似。虽然我们的研究仅限于二次模型，但我们认为它可以推广到任何顺序，并铺平了朝着强大而有效的策略进行严格理由以任意准确性近似水波的道理。我们的研究得到了详细且可重复的数值模拟的支持。
In this work we discuss an approximate model for the propagation of deep irrotational water waves, specifically the model obtained by keeping only quadratic nonlinearities in the water waves system under the Zakharov/Craig-Sulem formulation. We argue that the initial-value problem associated with this system is most likely ill-posed in finite regularity spaces, and that it explains the observation of spurious amplification of high-wavenumber modes in numerical simulations that were reported in the literature. This hypothesis has already been proposed by Ambrose, Bona, and Nicholls [4] but we identify a different instability mechanism. On the basis of this analysis, we show that the system can be "rectified". Indeed, by introducing appropriate regularizing operators, we can restore the well-posedness without sacrificing other desirable features such as a canonical Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient numerical integration. This provides a first rigorous justification for the common practice of applying filters in high-order spectral methods for the numerical approximation of surface gravity waves. While our study is restricted to a quadratic model, we believe it can be generalized to any order and paves the way towards the rigorous justification of a robust and efficient strategy to approximate water waves with arbitrary accuracy. Our study is supported by detailed and reproducible numerical simulations.
