A perturbation DRBEM model for weakly nonlinear wave run-ups around islands.

*(English)*Zbl 1188.76233Summary: The dual reciprocity boundary element method (DRBEM) based on the perturbation method is presented for calculating run-ups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations by using the perturbation method, where, besides nonlinearity \(\varepsilon\), the dispersion \(\mu^2\) is also included in the perturbed expansion. The first-order solution \(\eta _{0}\) is found by using the linear long-wave equations. Then \(\eta_0\) is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission and approximation for the seabed slope \(\nabla h\) and its derivatives is made, there are the third- and fourth-order partial derivatives of \(\eta_0\) appeared in the right-hand sides of two governing equations of the second-order. By employing a transformation, those third- and fourth-order partial derivatives are removed therefore large errors in approximating these derivatives are eliminated.

To validate the new model, wave diffractions around a large vertical cylinder for 13 cases are first considered. It is found that the nonlinear contributions to the new model are significant for weakly nonlinear waves with a much better comparison with experimental results obtained than for the linear diffraction theory. It is also found that the dispersive effects play an important role in improving the accuracy of the new model as numerical results obtained from the Boussinesq equations (with dispersion terms) are more accurate than those from the Airy’s equations (without dispersion term). Then the combined wave diffraction and refraction by a conical island is also modeled and discussed. Our model is not only accurate as the dispersive effects have been included but also computationally efficient since the domain integrals are merely evaluated by distributing collocation points over that surface.

To validate the new model, wave diffractions around a large vertical cylinder for 13 cases are first considered. It is found that the nonlinear contributions to the new model are significant for weakly nonlinear waves with a much better comparison with experimental results obtained than for the linear diffraction theory. It is also found that the dispersive effects play an important role in improving the accuracy of the new model as numerical results obtained from the Boussinesq equations (with dispersion terms) are more accurate than those from the Airy’s equations (without dispersion term). Then the combined wave diffraction and refraction by a conical island is also modeled and discussed. Our model is not only accurate as the dispersive effects have been included but also computationally efficient since the domain integrals are merely evaluated by distributing collocation points over that surface.

##### MSC:

76M15 | Boundary element methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

86A05 | Hydrology, hydrography, oceanography |

##### Keywords:

dual reciprocity boundary element method (DRBEM); nonlinear Boussinesq equations; wave diffraction and refraction
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\textit{S.-P. Zhu} et al., Eng. Anal. Bound. Elem. 33, No. 1, 63--76 (2009; Zbl 1188.76233)

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