This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet–Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M, and spatial resolution, N, and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.

中文翻译：

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一种新的隐式泰勒方法和高阶谱方法对非线性稳态波的精度和适用性

本文对隐式泰勒 (IT) 方法与经典高阶谱 (HOS) 方法在用于模拟二维规则波时的准确性和适用性进行了调查和讨论。这种比较是相关的，因为 HOS 方法实际上是 IT 公式的显式扰动解决方案。首先，我们考虑在给定表面高程和表面势的情况下确定自由表面垂直速度的 Dirichlet-Neumann 问题。对于这个问题，我们得出结论，当使用相同的截断阶数 M 和空间分辨率 N 时，IT 方法比 HOS 方法更准确，并且能够处理比 HOS 方法更陡峭的波浪。其次，我们关注的是两种方法在时间上的整合问题。在这方面，事实证明，对于类似的截断阶数，IT 方法不如 HOS 方法稳健。我们得出结论，IT 方法应限制为 M = 4，而 HOS 方法可用于 M ≤ 8。我们系统地比较了这两个选项，并最终确定这两种方法的最佳可实现精度作为波陡度的函数和水深。