A comparison of smoothed particle hydrodynamics simulation with exact results from a nonlinear water wave model

Improving performance of a semi-analytical model for nonlinear water waves

This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. At the heart of the numerical method is the solution to the Laplace equation which is solved using a variant of the ‐transform. The method discretizes the spatial part of the governing equations using the Galerkin method and the temporal part using the classical fourth‐order Runge‐Kutta method. A careful investigation of the numerical method's stability properties is carried out, and it is shown that the method is stable up to a certain threshold steepness when applied to nonlinear monochromatic waves in deep water. Above this threshold artificial damping may be employed to obtain stable solutions. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking.

Propagation of wave solutions of nonlinear Heisenberg ferromagnetic spin chain and Vakhnenko dynamical equations arising in nonlinear water wave models

The higher order of nonlinear partial differential equations in mathematical physics is studied. We used the analytical mathematical methods of the nonlinear (3+1)-dimensional extended Zakharov–Kuznetsov dynamical, modified KdV–Zakharov–Kuznetsov and generalized shallow water wave equations to demonstrate the efficiency and validity of the proposed powerful technique. The shallow water wave models have been applied in tidal waves and weather simulation. Exact wave solutions of these models in various forms such as Kink and anti-Kink solitons, bright–dark soliton, solitary wave and periodic solutions are constructed that have plenty of applications in diverse areas of physics. Graphically, we presented the movement of some obtained solitary wave solutions that aids in understanding the physical phenomena of these models.

We form the analytical solitary wave solutions with the execution of generalized direct algebraic technique on three well-known nonlinear wave models, namely, called foam drainage, longitudinal magnetoelectro-elastic circular rod and modified Degasperis–Procesi equations. The derived solutions are hyperbolic functions in which some are plotted graphically on meticulous values to the parameters which provides the basic knowledge to understand physical significant of these three wave models. The obtained solutions show the efficiency and precision of our scheme. These derived new results prove that our novel technique is awfully effective and can be productive as a instrument solving for sundry other nonlinear evolution equations.

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

Evolution equations for nonlinear long waves are considered from an approximation to the exact Hamiltonian (total energy) for the water waves. The approximation which is used here has two distinct advantages over many other formulations which are commonly used for the same purpose. Further, a variation of these evolution equations is considered in order to incorporate higher-order nonlinearity. Numerical solutions of the evolution equations have been carried out for both the systems. Application of these models is illustrated in some practical cases. Comparisons between experimental measurements and computed results show that the model can be used for satisfactory prediction of nonlinear transformation of non-breaking waves over varying depth. Two features for further investigation are: (i) inclusion of both short-wave and long-wave nonlinearity so that the model can be used with uniform validity from deep to shallow water and (ii) modifications of the evolution equations so that they can be applied to propagation of breaking waves in a robust way.

A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a formulation of a fully nonlinear and dispersive potential flow water wave model with random inputs for the probabilistic description of the evolution of waves. The model is analyzed using random sampling techniques and non-intrusive methods based on generalized Polynomial Chaos (PC). These methods allow to accurately and efficiently estimate the probability distribution of the solution and require only the computation of the solution in different points in the parameter space, allowing for the reuse of existing simulation software. The choice of the applied methods is driven by the number of uncertain input parameters and by the fact that finding the solution of the considered model is computationally intensive. We revisit experimental benchmarks often used for validation of deterministic water wave models. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in comparison with experimental measurements could be partially explained by the variability in the model input. We finally present a synthetic experiment studying the variance based sensitivity of the wave load on an off-shore structure to a number of input uncertainties. In the numerical examples presented the PC methods have exhibited fast convergence, suggesting that the problem is amenable to being analyzed with such methods.

An efficient flexible-order model for 3D nonlinear water waves

Fully dispersive nonlinear water wave model in curvilinear coordinates

In this paper, our main aim is to find the new travelling wave solutions to the Wu–Zhang system which describes dispersive long waves. Some new and important travelling wave solutions with rational and exponential function structures are successfully constructed by using the improved Bernoulli subequation function method. All the obtained solutions in this study satisfy the Wu–Zhang system. The interesting three- and two-dimensional surfaces to all the obtained solutions are plotted. We carried out all the computations and the graphics plots in this paper with aid of the Wolfram Mathematica 9.

This paper presents a weakly nonlinear water wave model using a mild slope equation and a new explicit formulation which takes into account dispersion of wave phase velocity, approximates Hedges' (1987) nonlinear dispersion relationship, and accords well with the original empirical formula. Comparison of the calculating results with those obtained from the experimental data and those obtained from linear wave theory showed that the present water wave model considering the dispersion of phase velocity is rational and in good agreement with experiment data.

Nonlinear wave generation by a wavemaker in deep to intermediate water depth

Investigation of Bragg resonant reflection of linear and nonlinear water waves by multiple fixed floating structures based on viscous meshfree method

This article investigates the traveling wave solution of the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model by using the complete discriminant system method. These solutions not only include rational function solutions, trigonometric function solutions, but also Jacobian function solutions. In order to illustrate the propagation of these solutions in the field of nonlinear optics and water wave models, some three-dimensional, two-dimensional, and contour maps are drawn. Meanwhile, the phase portrait of two-dimensional dynamical systems and its perturbation systems are studied using the planar dynamical system analysis method. By drawing phase diagrams, it is easy to observe the stability, periodicity, and chaotic behavior of two-dimensional dynamical systems through geometric visualization, which can also provide strong basis for researchers to design corresponding control systems.