High-order strongly nonlinear long wave approximation and solitary wave solution
High-order strongly nonlinear long wave approximation and solitary wave solution
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DOI：
10.1017/jfm.2022.544
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发表时间：
2022-07
影响因子：
3.7
通讯作者：
W. Choi
中科院分区：
文献类型：
--
作者：
W. Choi
摘要 我们考虑将高阶强非线性长波模型扩展为单个小参数，测量水深与特征波长的比率。通过检查其色散关系，发现底部速度的高阶系统对于任何近似阶数的所有扰动都是稳定的。另一方面，其他速度的系统可能不稳定，甚至不适定，正如无限的最大增长所表明的那样。在稳定假设下，利用高阶强非线性系统得到了欧拉方程组的新的三阶孤立波解，并在振幅参数上展开，这与弱非线性理论中使用的不同。三阶解可以很好地描述有限振幅孤立波引起的各种物理量，包括波廓、水平速度廓线、波峰粒子速度和波底压力。对于数值计算，考虑底部速度的一阶和二阶强非线性系统。结果表明，有限差分格式由于在近似高阶空间导数时引入的截断误差而不稳定，因此需要更精确的空间离散化格式。采用基于有限傅里叶级数的伪谱方法结合非局部算子反演的迭代方案，对强非线性系统中单个孤立波的传播和两个反向传播的有限振幅孤立波的正面碰撞进行了数值求解，并将结果与以前的实验室测量结果进行了比较。
Abstract We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.