Numerical simulation for the two-dimensional nonlinear shallow water waves

This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory.

Painlevé integrability and a collection of new wave structures related to an important model in shallow water waves

This paper investigates the perturbed Boussinesq equation that emerges in shallow water waves.The perturbed Boussinesq equation describes the properties of longitudinal waves in bars,long water waves,plasma waves,quantum mechanics,acoustic waves,nonlinear optics,and other phenomena.As a result,the governing model has significant importance in its own right.The singular manifold method and the unified methods are employed in the proposed model for extracting hyperbolic,trigonometric,and rational function solutions.These solutions may be useful in determining the underlying context of the physical incidents.It is worth noting that the executed methods are skilled and effective for examining nonlinear evaluation equations,compatible with computer algebra,and provide a wide range of wave solutions.In addition to this,the Painlevétest is also used to check the integrability of the governing model.Two-dimensional and threedimensional plots are made to illustrate the physical behavior of the newly obtained exact solutions.This makes the study of exact solutions to other nonlinear evaluation equations using the singular manifold method and unified technique prospective and deserving of further study.

Bilinear forms through the binary Bell polynomials, N solitons and Backlund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.

Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters

In this study,the potential Kadomtsev-Petviashvili(pKP)equation,which describes the oblique interaction of surface waves in shallow waters,is solved by the new extended direct algebraic method.The results of the study show that by applying the new direct algebraic method to the pKP equation,the behavior of the obliquely interacting surface waves in two dimensions can be analyzed.This article fairly clarifies the behaviors of surface waves in shallow waters.In the literature,several mathematical models have been developed in attempt to study these behaviors,with nonlinear mathematics being one of the most important steps;however,the investigations are still at a level that can be called‘baby steps’.Therefore,every study to be carried out in this context is of great importance.Thus,this study will serve as a reference to guide scientists working in this field.

Surface Gravity Waves: Shallow Water Fluid Particle Physics

One of Newton’s mathematical solutions to a hypothetical orbital problem, recently verified by an independent physics model, is applied to the fluid particle motion in shallow water surface gravity waves. What is the functional form of the central force, with origin at the ellipse’s center, which will keep a body in the orbit? Newton found out it is the spring force, which is linear. All fluid particles in shallow water waves move in ellipses. By a superposition of solutions in a linear problem, the application of Newton’s result to shallow water waves is combined with a feature not noticed by Newton: the orbital period is independent of the semi-major and semi-minor axes. Two conclusions reached are that the wave period of shoaling waves should be constant and that there is no friction in these waves.

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

Collisions Between Lumps/Rogue Waves and Solitons for A(3+1)-Dimensional Generalized Variable-Coefficient Shallow Water Wave Equation

In this paper,we investigate a(3+1)-dimensional generalized variable-coefficient shallow water wave equation,which can be used to describe the flow below a pressure surface in oceanography and atmospheric science.Employing the Kadomtsev−Petviashvili hierarchy reduction,we obtain the semi-rational solutions which describe the lumps and rogue waves interacting with the kink solitons.We find that the lump appears from one kink soliton and fuses into the other on the x−y and x−t planes.However,on the x−z plane,the localized waves in the middle of the parallel kink solitons are in two forms:lumps and line rogue waves.The effects of the variable coefficients on the two forms are discussed.The dispersion coefficient influences the speed of solitons,while the background coefficient influences the background’s height.

Combined refraction and diffraction models for sea waves over complicated topography and with currents in shallow water

-Combined refraction and diffraction models in the form of linear parabolic approximation are derived through smallparameter method. More strictly theoretical basis and more accuracy in the models than Lozano's (1980) are obtained. Some theoretical defects in Liu's model (1985) with consideration of current are not only found but also eliminated. More strict and accurate models are, therefore, presented in this paper.The calculation results and analysis in applying the models to actual wave field with consideration of bottom friction will be given in the following paper.

Modified (2+1)-dimensional displacement shallow water wave system and its approximate similarity solutions

Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.

Symmetries and Similarity Reductions of a New (2+1)-Dimensional Shallow Water Wave System

The symmetries of a （2＋1）-dimensional shallow water wave system, which is newly constructed through applying variation principle of analytic mechanics, are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. The （1＋1） dimensional displacement shallow water wave equation can be derived from the reductions when special symmetry parameters are chosen.

Kinetic Flux Vector Splitting Method for the Shallow Water Wave Equations

Based on the analogy to gas dynamics, the kinetic flux vector splitting (KFVS) method is used to stimulate the shallow water wave equations. The flux vectors of the equations are split on the basis of the local equilibrium Maxwell-Boltzmann distribution. One dimensional examples including a dam breaking wave and flows over a ridge are calculated. The solutions exhibit second-order accuracy with no spurious oscillation.

Transformation of Wave Energy Spectrum in Shallow Water

Combined with irregular wave-maker, the growing process of Wave Energy Spectrum in shallow water can be studied in wind wave channel on different water depth conditions, and its transformation characteristics and rules can be obtained.

An adaptive artificial viscosity for the displacement shallow water wave equation

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom,the moving wet-dry interface,and so on.In this paper,an adaptive artificial viscosity(AAV)is proposed and combined with the displacement shallow water wave equation(DSWWE)to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface.The effectiveness of the proposed AAV is first illustrated by using the steady-state solution and the small perturbation analysis.Then,the action mechanism of the AAV on the shallow water waves with the uneven bottom is explained by using the Fourier theory.It is shown that the AVV can suppress the wave with the large wave number,and can also suppress the numerical oscillations for the rapidly varying bottom.Finally,four numerical examples are given,and the numerical results show that the DSWWE combined with the AAV can effectively simulate the shock waves,accurately capture the movements of wet-dry interfaces,and precisely preserve the mass.

Applications of sediment sudden deposition model based on the third-generation numerical model for shallow water wave

The existing numerical models for nearshore waves are briefly introduced, and the third-generation numerical model for shallow water wave, which makes use of the most advanced productions of wave research and has been adapted well to be used in the environment of seacoast, lake and estuary area, is particularly discussed. The applied model realizes the significant wave height distribution at different wind directions. To integrate the model into the coastal area sediment, sudden deposition mechanism, the distribution of average silt content and the change of sediment sudden deposition thickness over time in the nearshore area are simulated. The academic productions can give some theoretical guidance to the applications of sediment sudden deposition mechanism for stormy waves in the coastal area. And the advancing directions of sediment sudden deposition model are prospected.

THE DESIGN OF CHARACTERIZING-INTEGRAI SCHEMES AND THE APPLICATION TO THE SHALLOW WATER WAVE PROBLEMS

In this paper a new approach for designing upwind type schemes-the characterizing-integral method and its applied skills are introduced. The method is simple, convenient and eff ective. And the method isn 't only limited to conservation laws unlike other methods and maybe easily extended to multi-dimension problems. Furthermore, the numerical dissipation of the method can be flexibly regulated, so that it is especially suitable for solving various discontinuity problems.The paper shows us now to use this approach to simulate deformation and breaking of a nonlinear shallow water wave on a gentle slope, and to compute two-dimensional dam failure problem.

Some New Nonlinear Wave Solutions for a Higher-Dimensional Shallow Water Wave Equation

In this manuscript, we first perform a complete Lie symmetry classification for a higher-dimensional shallow water wave equation and then construct the corresponding reduced equations with the obtained Lie symmetries. Moreover, with the extended F-expansion method, we obtain several new nonlinear wave solutions involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function and trigonometric function.

Explicit Kinetic Flux Vector Splitting Scheme for the 2-D Shallow Water Wave Equations

Originally, the kinetic flux vector splitting (KFVS) scheme was developed as a numerical method to solve gas dynamic problems. The main idea in the approach is to construct the flux based on the microscopical description of the gas. In this paper, based on the analogy between the shallow water wave equations and the gas dynamic equations, we develop an explicit KFVS method for simulating the shallow water wave equations. A 1D steady flow and a 2D unsteady flow are presented to show the robust and accuracy of the KFVS scheme.

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size （2a）, storm speed （U）, and beach slope （a）. Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, Ucr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 〈 y 〈 Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.

A THIRD-ORDER BOUSSINESQ MODEL APPLIED TO NONLINEAR EVOLUTION OF SHALLOW-WATER WAVES

The conventional Boussinesq model is extended to the third order in dispersion and nonlinearity. The new equations are shown to possess better linear dispersion characteristics. For the evolution of periodic waves over a constant depth, the computed wave envelops are spatially aperiodic and skew. The model is then applied to the study of wave focusing by a topographical lens and the results are compared with Whalin′s (1971) experimental data as well as some previous results from the conventional Boussinesq model. Encouragingly, improved agreement with Whalin′s experimental data is found. [WT5”HZ]

NUMERICAL MODELING OF WAVE EVOLUTION AND RUNUP IN SHALLOW WATER

Based on the Navier-Stokes （N-S） equations for viscous, incompressible fluid and the VOF method, 2-D and 3-D Numerical Wave Tanks （NWT） for nonlinear shallow water waves are built. The dynamic mesh technique is applied, which can save computational resources dramatically for the simulation of solitary wave propagating at a constant depth. Higher order approximation for cnoidal wave is employed to generate high quality waves. Shoaling and breaking of solitary waves over different slopes are simulated and analyzed systematically. Wave runup on structures is also investigated. The results agree very well with experimental data or analytical solutions.