基于蛙跳格式的弱非线性水波数值模型

A weakly nonlinear model for water waves using frog-loop scheme

为了较好地模拟非线性波浪传播变形,采用Beji和Nadaoka(1996)的弱非线性Boussinesq水波方程,在非交错网格下建立数值模型.数值计算中,模型中时间导数项采用蛙跳格式,空间一次导数项采用采用5点4阶精度格式,其他导数项采用2阶精度格式.利用模型计算了潜堤、椭圆型浅滩和凹型浅滩三个地形上的波浪传播变形,再现了波浪传播中的变浅、折射、绕射以及反射等现象.潜堤算例比较了测量点处的波面时间历程,椭圆型浅滩算例比较了均方波高值,凹型浅滩算例比较了沿中心线的各次谐波值,模拟结果与相应的实验值的比较可见吻合程度较好.这说明蛙跳格式可以用于数值求解弱非线性的Boussinesq水波方程,其数值模拟效果合理,可用于波浪场的实际模拟.

To better model the nonlinear wave propagation and deformation, the extended Boussinesq model with weak nonlinearity was used, and the related numerical model was established in non -staggered grids. In the numerical simulations, the frog - loop scheme was used for time stepping. Terms involved first - order spatial derivatives were differential to fourth -order accuracy with five -point formula, and the other derivatives to second -order accuracy. Numerical simulations were done upon wave evolution over three typical topographies： submerged sill, elliptic shoal and concave type shoal, and the phenomena such as shoaling, refraction, diffraction and reflection were presented numerically. For the submerged sill example, the surface elevation varying with the time at fix locations was compared; for the elliptic shoal example, the mean square wave height was compared; and for the concave type shoal, harmonics wave amplitude was compared along center line. The good agreement between related numerical results and experimental results shows that the frog -loop scheme can be used to solve the extended Boussinesq equations with weak nonlinearity effectively, and this model can be used to simulate the wave field in reality.