In designing a numerical tool for simulating a wide variety of water waves, i.e. short to long waves, an accurate and robust wave model and numerical implementation are needed. Dispersion and nonlinearity are the two most important physical aspects that should be modeled accurately. To be applicable to simulate many coastal engineering applications, the numerical scheme should be capable of simulating wave runup and overtopping. In this paper, we extend the capability of a Boussinesq-type model called Variational Boussinesq (VB) model for simulating the runup and overtopping of water waves. To that end, the vertical layer of the fluid is modeled continuously by a linear combination of three functions. If two of these three functions have been incorporated in the previous numerical approximation called the SVB model, this paper discusses the improvement of SVB model by incorporating all the three functions. This approach improve the dispersive property of the SVB model due to its ability to simulate short waves up to kd = 20, compared to the previous model which was only up to kd = 7, where k denotes wave number and d water depth. Furthermore, the model is implemented numerically by using the staggered conservative scheme. In the new implementation, the model is switched to the non-dispersive Shallow Water Equations (SWE) when dealing with a dry area for runup and overtopping phenomena. The new implementation is tested against analytical solutions of soliton propagation and standing wave phenomenon; moreover, it is also tested against experimental data from hydrodynamic laboratories for simulating solitary wave breaking above a sloping bottom, composite beach, and in a structure for simulating overtopping phenomenon. The implementation is also tested against experimental data for simulating irregular wave propagation and runup above a fringing reef. The results of numerical simulation agree quite well with experimental data.

中文翻译：

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使用交错网格变分 Boussinesq 数值模拟短波和长波的浪涌和越浪

在设计用于模拟各种水波（即短波到长波）的数值工具时，需要准确且稳健的波浪模型和数值实现。色散和非线性是应该准确建模的两个最重要的物理方面。为了适用于模拟许多海岸工程应用，数值方案应该能够模拟波浪的上升和溢出。在本文中，我们扩展了称为 Variational Boussinesq (VB) 模型的 Boussinesq 型模型的能力，用于模拟水波的上升和越浪。为此，流体的垂直层通过三个函数的线性组合连续建模。如果这三个函数中的两个已被合并到前面称为 SVB 模型的数值近似中，本文讨论了通过结合所有三个功能对 SVB 模型的改进。这种方法改善了 SVB 模型的色散特性，因为它能够模拟高达 kd = 20 的短波，而之前的模型仅高达 kd = 7，其中 k 表示波数和 d 水深。此外，该模型通过使用交错保守方案在数值上实现。在新的实施中，在处理干燥区域的上升和溢出现象时，模型切换到非色散浅水方程 (SWE)。新实现针对孤子传播和驻波现象的解析解进行了测试；此外，它还针对来自水动力实验室的实验数据进行了测试，以模拟倾斜底部、复合海滩上方的孤立波破裂，以及用于模拟越界现象的结构。该实现还针对模拟不规则波传播和在边缘礁上方的上升的实验数据进行了测试。数值模拟结果与实验数据吻合较好。