*2.2. High-Order Spectral (HOS) Method*

Combined with the advantages of Zakharov equations and mode-coupling ideas based on potential theory (with the application of the Fast Fourier Transform), the HOS method has been proved as a suitable, robust, and highly efficient numerical method for direct phase-resolved simulation of nonlinear ocean wave field evolution [46–48], wave nonlinear mechanism analysis [49–52] and nonlinear wave-wave interactions and wavebody interactions [31,53,54]. In this paper, the HOS numerical model for three-dimensional Bragg resonance is considered based on the HOS method for general nonlinear wavebottom interactions developed by Liu [20]. In addition, the focusing properties of the three-dimensional wave field caused by Bragg resonance with waves passing through V-shaped undulating bottom are studied.

The difficulty of establishing a numerical model is to deal with free-surface boundary conditions. To simplify the calculation, the high-order spectral (HOS) method introduces the velocity potential of the free surface in reference to Zakharov theory [55], so the freesurface boundary condition can be written in the following form:

$$\begin{aligned} \eta\_t + \nabla\_x \eta \cdot \nabla\_x \phi^s - (1 + \nabla\_x \eta \cdot \nabla\_x \eta) \phi\_\sharp(\mathbf{x}, \eta, t) &= 0 \\ \phi^s + \mathbf{g}\eta + \frac{1}{2} \nabla\_x \phi^s \cdot \nabla\_x \phi^s - \frac{1}{2} (1 + \nabla\_x \eta \cdot \nabla\_x \eta) \phi\_\sharp^{-2}(\mathbf{x}, \eta, t) &= 0 \end{aligned} \tag{3}$$
