Water Wave Model Research Articles

The existence of solitary wave solutions of delayed Camassa–Holm equation via a geometric approach

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation.

Numerical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

In this paper, we numerically study the water wave model with a nonlocal viscous term urn:x-wiley:mma:media:mma4932:mma4932-math-0001 where is the Riemann‐Liouville half‐order derivative in time. We propose and compare different numerical schemes based on the diffusive realizations of fractional operators.

Isobe–Kakinuma model for water waves as a higher order shallow water approximation

We justify rigorously an Isobe–Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order O(δ2), where δ is a small nondimensional parameter defined as the ratio of the mean depth to the typical wavelength. The Green–Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order O(δ4). In this paper we show that the Isobe–Kakinuma model is a much higher order approximation to the water wave equations with an error of order O(δ6).

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

Time two-mesh algorithm combined with finite element method for time fractional water wave model

In this article, a new time two-mesh (TT-M) finite element (FE) method, which is constructed by a new TT-M algorithm and FE method in space, is proposed and analyzed. The numerical theories and algorithm are shown by solving the fractional water wave model including fractional derivative in time. The TT-M FE algorithm mainly covers three steps: firstly, a nonlinear FE system at some time points based on the time coarse mesh ΔtC is solved by an iterative method; further, based on the obtained numerical solution on time coarse mesh ΔtC in the first step, some useful numerical solutions between two time coarse mesh points are arrived at by the Lagrange’s interpolation formula; finally, the solutions on the first and second steps are chosen as the initial iteration value, then a linear FE system on time fine mesh ΔtF<ΔtC is solved. Some stable results and a priori error estimates are analyzed in detail. Furthermore, some numerical results are provided to verify the effectiveness of TT-M FE method. By the comparison with the standard FE method, it is easy to see that the CPU time can be saved by our TT-M FE method.

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document} , \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document} , and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

Soliton solutions for (2+1) and (3+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony model equations and their applications

The Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) model equations as a water wave model, are governing equations, for fluid flows, describes bidirectional propagating water wave surface. The soliton solutions for (2+1) and (3+1)-Dimensional Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equations have been extracted. The solitary wave ansatz method are adopted to approximate the solutions. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of the problem.

Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

In this paper, we review the history and current state-of-the-art in the modelling of long nonlinear dispersive waves. For the sake of conciseness of this review, we omit the unidirectional models and focus especially on some classical and improved Boussinesq-type and Serre-Green-Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

Dispersive Shallow Water Wave Modelling. Part III: Model Derivation on a Globally Spherical Geometry

The present article is the third part of a series of papers devoted to the shallow water wave modelling. In this part, we investigate the derivation of some long wave models on a deformed sphere. We propose first a suitable for our purposes formulation of the full Euler equations on a sphere. Then, by applying the depth-averaging procedure we derive first a new fully nonlinear weakly dispersive base model. After this step, we show how to obtain some weakly nonlinear models on the sphere in the so-called Boussinesq regime. We have to say that the proposed base model contains an additional velocity variable which has to be specified by a closure relation. Physically, it represents a dispersive correction to the velocity vector. So, the main outcome of our article should be rather considered as a whole family of long wave models.

Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

In this paper, we describe a numerical method to solve numerically the weakly dispersive fully nonlinear Serre-Green-Naghdi (SGN) celebrated model. Namely, our scheme is based on reliable finite volume methods, proven to be very effective for the hyperbolic part of equations. The particularity of our study is that we develop an adaptive numerical model using moving grids. Moreover, we use a special form of the SGN equations where non-hydrostatic part of pressure is found by solving a nonlinear elliptic equation. Moreover, this form of governing equations allows determining the natural form of boundary conditions to obtain a well-posed (numerical) problem.

On the existence and computation of periodic travelling waves for a 2D water wave model

In this work we establish the existence of at least one weak periodic solution in the spatial directions of a nonlinear system of two coupled differential equations associated with a 2D Boussinesq model which describes the evolution of long water waves with small amplitude under the effect of surface tension. For wave speed \begin{document} $0 , the problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space \begin{document} $H^{1}_k(\mathbb{R})$ \end{document} ( \begin{document} $k$ \end{document} -periodic functions \begin{document} $f∈ L_k^2(\mathbb{R})$ \end{document} such that \begin{document} $f' ∈ L_k^2(\mathbb{R})$ \end{document} ). For wave speed \begin{document} $|c|>1$ \end{document} , the result is a direct consequence of the Lyapunov Center Theorem since the nonlinear system can be rewritten as a \begin{document} $4× 4$ \end{document} system with a special Hamiltonian structure. In the case \begin{document} $|c|>1$ \end{document} , we also compute numerical approximations of these travelling waves by using a Fourier spectral discretization of the corresponding 1D travelling wave equations and a Newton-type iteration.

Dispersive Shallow Water Wave Modelling. Part IV: Numerical Simulation on a Globally Spherical Geometry

In the present manuscript, we consider the problem of dispersive wave simulation on a rotating globally spherical geometry. In this Part IV, we focus on numerical aspects while the model derivation was described in Part III. The algorithm we propose is based on the splitting approach. Namely, equations are decomposed on a uniformly elliptic equation for the dispersive pressure component and a hyperbolic part of shallow water equations (on a sphere) with source terms. This algorithm is implemented as a two-step predictor-corrector scheme. On every step, we solve separately elliptic and hyperbolic problems. Then, the performance of this algorithm is illustrated on model idealised situations with an even bottom, where we estimate the influence of sphericity and rotation effects on dispersive wave propagation. The dispersive effects are quantified depending on the propagation distance over the sphere and on the linear extent of generation region. Finally, the numerical method is applied to a couple of real-world events. Namely, we undertake simulations of the Bulgarian 2007 and Chilean 2010 tsunamis. Whenever the data is available, our computational results are confronted with real measurements.

On Whitham and Related Equations

AbstractThe aim of this paper is to study, via theoretical analysis and numerical simulations, the dynamics of Whitham and related equations. In particular, we establish rigorous bounds between solutions of the Whitham and Korteweg–de Vries equations and provide some insights into the dynamics of the Whitham equation in different regimes, some of them being outside the range of validity of the Whitham equation as a water waves model.

On the Mathematical Description of Time-Dependent Surface Water Waves

This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and global well-posedness results for the model equations. Moreover, we present reduced models in various parameter regimes for the approximate description of the motion of typical wave profiles and discuss the mathematically rigorous justification of the validity of these models.

Variational modelling of wave\u2013structure interactions with an offshore wind-turbine mast

We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. We develop a linearized model of the fluid–structure or wave–mast coupling, which is a linearization of the variational principle for the fully coupled nonlinear model. Our numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The energy exchange between the subsystems is seen to be in balance, yielding a total energy that shows only small and bounded oscillations amplitude of which tends to zero with the second-order convergence as the timestep approaches zero. Similar second-order convergence is observed for spatial mesh refinement. The linearized model so far developed can be extended to a nonlinear regime.

Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves

We consider the initial value problem to the Isobe-Kakinuma model for water waves and the structure of the model. The Isobe-Kakinuma model is the Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The Isobe-Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations. Since the hypersurface $t=0$ is characteristic for the Isobe-Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces. We also discuss the linear dispersion relation to the model.

Topological soliton solutions for three shallow water waves models

In this article, we investigate three distinct physical structures for shallow water waves models by the improved ansatz method. The method was improved and can be used to obtain more generalized form topological soliton solutions than the original method. As a result, some new exact solutions of the shallow water equations are successfully established and the obtained results are exhibited graphically. The results showed that the improved ansatz method can be applied to solve other nonlinear differential equations arising from mathematical physics.

Abundant Solutions of Distinct Physical Structures for Three Shallow Water Waves Models

Abundant Solutions of Distinct Physical Structures for Three Shallow Water Waves Models

Assessment of URANS surface effect ship models for calm water and head waves

Surface effect ship (SES) air cushion and seal models are implemented in an URANS hydrodynamics solver. The air cushion is modeled either as a prescribed pressure patch, or as a compressible isothermal/adiabatic ideal stagnant air with fan and leakage flows. The seals are either discretized as hinged bodies or modeled as 2D planing surfaces with hydrodynamic interaction. Verification and validation studies are performed using T-Craft experimental data for calm water resistance, sinkage and trim at Froude number (Fr)=0.1–0.6; impulsive heave and pitch decay at Fr=0; and wave-induced resistance and motion predictions in head waves at Fr=0 and 0.6. The compressible air cushion model with fan and leakage flows perform better than those without the fan and leakage flows and the prescribed pressure patch model. The hinged seal model performs better than the 2D planing surface model, but is computationally expensive for time accurate simulations. Therefore, the 2D planing surface model is used for the validation studies. SES simulations on grids with 5.3M cells show grid verification intervals of 6%, which are comparable to those reported for displacement and semi-planing hull studies on similar grid sizes. On an average calm water and impulsive motion predictions compare within 8.5% of the experimental data, and wave-induced motion predictions show somewhat larger error of 13.5%. The errors levels are mostly comparable to those for displacement and semi-planing and planing hulls. The study identifies that most critical advancement needed for SES simulations is the seal modeling including fluid structure interaction.

Numerical scheme for a model of shallow water waves in [formula omitted]-dimensions

A nonlinear difference scheme is considered for the two-dimensional Rosenau–Burgers equation. Some priori estimates, existence and uniqueness of the difference solution have been shown. A second-order convergence in the uniform norm and stability are proved. Also a convergent iterative algorithm is presented. All results are obtained without any restrictions on the meshsizes. At last numerical experiments are carried out to support the theoretical claims.