Dispersive Shallow Water Wave Research Articles

The Efficient Method to Solve the Conformable Time Fractional Benney Equation

The current study employed the innovative conformable fractional method to analyze the nonlinear Benney equations involving the conformable fractional derivative. Conformable fractional Benney equations have been examined by the conformable q‐Shehu analysis transform method. By including nonlinear factors, it offers a more precise depiction of wave propagation compared to linear models. Various natural phenomena, including ocean waves, plasma waves, and some forms of solitons, display nonlinear behavior that cannot be precisely explained by linear equations. The fractional Benney equation is important because it extends the classical Benney equation, which describes the evolution of weakly nonlinear and weakly dispersive long waves in shallow water. By incorporating fractional calculus operators, the fractional Benney equation provides a more accurate description of wave propagation phenomena in certain physical systems characterized by nonlocal or memory‐dependent behavior. The utilization of the Benney equation enables researchers to simulate these occurrences with greater realism. This study investigates the convergence and inaccuracy of the future scheme. The conformable q‐Shehu homotopy analysis transform method (Cq‐SHATM) generates h‐curves that demonstrate the convergence interval of the series solution obtained. In order to determine the effectiveness and suitability of the Cq‐SHATM, uniqueness and convergence theorems have been proven. This study presents an application that showcases the potential advantages and efficacy of the suggested method. Moreover, an error analysis is conducted to validate the precision of the scheme. Computational simulations are performed to verify the accuracy of the upcoming method. This study presents the results gained from the numerical and graphical analysis. The method presented in this work demonstrates a high level of computational accuracy and simplicity in analyzing and solving complex phenomena associated with conformable fractional nonlinear partial differential equations in the fields of science and technology.

Oceanic shallow-water investigations on a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system

To date, with respect to water waves, researchers have studied certain systems, including a generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system that describes, e.g., the dispersive long waves in the oceanic shallow water, which we study here. With respect to, e.g., the horizontal velocity of the water wave as well as the height of the deviation from the equilibrium position of the water, with symbolic computation, on the one hand, the system is found to pass the Painlevé test under some coefficient constraints, while on the other hand, two families of the bilinear forms and two families of the N-soliton solutions are constructed, with N as a positive integer. Related constraints are shown. Our bilinear forms and N-soliton solutions depend on the coefficients in the system.

Reconstructions of time-evolving sound-speed fields perturbed by deformed and dispersive internal solitary waves in shallow water

The high-fidelity reconstruction of sound speeds is crucial for predicting acoustic propagation in shallow water where internal solitary waves (ISWs) are prevalent. Mapping temperatures from time series to spatial fields is an approach widely used to reproduce the sound speed perturbed by deformed internal waves. However, wave-shape distortions are inherent in the modeling results. This paper analyzes the formation mechanism and dynamic behavior of the distorted waveform that is shown to arise from the mismatch between the modeled and real propagation speeds of individual solitons within an ISW packet. To mitigate distortions, a reconstruction method incorporating the dispersion property of an ISW train is proposed here. The principle is to assign each soliton a real speed observed in the experiment. Then, the modeled solitons propagate at their intrinsic speeds, and the packet disperses naturally with time. The method is applied to reconstruct the sound speed perturbed by ISWs in the South China Sea. The mean and median of the root-mean-square error between the reconstructed and measured sound speeds are below 2 m/s. The modeled shape deformations and packet dispersion agree well with observations, and the waveform distortion is reduced compared with the original method. This work ensures the high fidelity of waveguide-environment reconstructions and facilitates the investigation of sound propagation in the future.

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

On a Whitham-Broer-Kaup-like system arising in the oceanic shallow water

Presently, taking care of the oceanic shallow water, people have investigated some Kaup-type systems. For example, a Whitham-Broer-Kaup-like system, which we aim to study hereby, models the dispersive long waves in the oceanic shallow water. As for the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, using symbolic computation, we build up (1) a scaling transformation, (2) two hetero-Bäcklund transformations, each of which is from that system to a known linear partial differential equation, and (3) a set of the similarity reductions, from that system to a known ordinary differential equation with some sample solutions also presented. Our results rely on different dispersion/diffusion powers for that system, with respect to the oceanic shallow water.

Numerical approach of dispersive shallow water waves with Rosenau-KdV-RLW equation in (2 + 1)-dimensions

Numerical approach of dispersive shallow water waves with Rosenau-KdV-RLW equation in (2 + 1)-dimensions

Thinking about the oceanic shallow water via a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system

Currently, fluid mechanics has been paid attention to. Hereby, making use of symbolic computation, we investigate a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system describing, e.g., the dispersive long waves in the oceanic shallow water. As for, e.g., the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, we work out (1) two sets of the hetero-Bäcklund transformations, each of which, from that system to a known linear partial differential equation, and (2) two sets of the similarity reductions, each of which, from that system to a known ordinary differential equation. Our hetero-Bäcklund transformations and similarity reductions depend on the coefficients in that system, as for, e.g., the oceanic shallow water.

Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity

This paper studies the asymptotic behavior of a solution of dispersive shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation. The energy decay rates of the solution to the model are examined through the Fourier transform method. Moreover, we develop a pseudo-compact finite difference scheme for solving the model, study solution behavior, and confirm our theoretical results. The fundamental energy-decreasing property, which is obtained from the model of coupling the Rosenau-RLW equation with the Rosenau-Burgers equation, is derived and preserved by the present numerical scheme. The existence, uniqueness, convergence, and stability of the numerical solution are theoretically analyzed. Some numerical experiments are also conducted to demonstrate the accuracy and robustness of the present method. Finally, to verify the optimal decay rates, the numerical results are carried out at variant time scales by applying the moving boundary technique. The simulations are successfully constructed to support the theoretical results.

Approximate Numerical solutions for the nonlinear dispersive shallow water waves as the Fornberg–Whitham model equations

The nonlinear partial differential equations having travelling or solitary wave solutions is numerically challenging, in which one of the important type is the Fornberg–Whitham model equation. This article aims to solve the Fornberg–Whitham type equations numerically via the variational iteration algorithm-I (MVIA-I). The MVIA-I gives approximate and exact solutions with easily computable terms to linear and nonlinear PDEs without the linearization or discretization, small perturbation and Adomian polynomials. To assess the precision, reliability and compactness of the recommended algorithm, we have compared the obtained results with the traditional variational iteration method (VIM), homotopy analysis method, reproducing kernel Hilbert space method and Adomian’s decomposition method which reveals that the MVIA-I is computationally attractive, exceptionally productive and is more reliable than the others techniques used in the literature.

Long waves in oceanic shallow water: Symbolic computation on the bilinear forms and Bäcklund transformations for the Whitham–Broer–Kaup system

For the Earth, water is at the core of sustainable development and at the heart of adaptation to climate change. For the Enceladus, the sixth-largest moon of the Saturn, Cassini spacecraft discovers a possible global ocean of liquid water beneath an icy crust. Oceanic water waves are one of the most common phenomena in nature. Hereby, on the Whitham–Broer–Kaup system for the dispersive long waves in the oceanic shallow water, with respect to the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, our binary Bell polynomials and symbolic computation lead to two sets of the bilinear forms, two sets of the N-soliton solutions and two sets of the auto-Backlund transformations with the sample solitons, where N is a positive integer. Our bilinear forms and auto-Backlund transformations are different from those reported in the existing literatures. All of our results are dependent on the coefficients in the system which represent different oceanic-water-wave dispersion/diffusion powers.

An efficient computational approach for two-dimensional variant of nonlinear-dispersive model of shallow water wave

A three-level linearized difference scheme for two-dimensional dispersive shallow water wave that is governed by the Rosenau-RLW equation is considered. It is proved that the proposed difference scheme is conservative, uniquely solvable and unconditionally convergent. The convergence order in maximum norm is $$O(\tau ^2+h_1^2+h_2^2)$$, where $$\tau$$ is the temporal grid size and $$h_1, h_2$$ are spatial grid sizes in the x- and y-directions, respectively. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and other existing method. Comparison reveals that our method improves the accuracy of the space and time direction and shortens computation time largely.

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.

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.

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.

A New Functional Iterative Algorithm for the Regularized Long-Wave Equation Using an Integral Equation Formalism

A fundamental question in computational nonlinear partial differential equations is raised to discover if one could construct a functional iterative algorithm for the regularized long-wave (RLW) equation (or the Benjamin–Bona–Mahony equation) based on an integral equation formalism? Here, the RLW equation is a third-order nonlinear partial differential equation, describing physically nonlinear dispersive waves in shallow water. For the question, the concept of pseudo-parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118–138, 2017), is introduced and incorporated into the RLW equation. Thereby, dual nonlinear integral equations of second kind involving the parameter are formulated. The application of the fixed point theorem to the integral equations results in a new (semi-analytic and derivative-free) functional iteration algorithm (as required). The new algorithm allows the exploration of new regimes of pseudo-parameters, so that it can be valid for a much wider range (in the complex plane) of pseudo-parameter values than that of Jang (2017). Being fairly simple (or straightforward), the iteration algorithm is found to be not only stable but accurate. Specifically, a numerical experiment on a solitary wave is performed on the convergence and accuracy of the iteration for various complex values of the pseudo-parameters, further providing the regions of convergence subject to some constraints in the complex plane. Moreover, the algorithm yields a particularly relevant physical investigation of the nonlinear behavior near the front of a slowly varying wave train, in which, indeed, interesting nonlinear wave features are demonstrated. As a consequence, the preceding question may be answered.

Wavelet Galerkin scheme for solving nonlinear dispersive shallow water waves: Application in bore propagation and breaking

An accurate and fast numerical method is developed to investigate the nonlinear (linear) shallow water wave propagation over flat (depth-varying) topography in one space dimension within an irrotational and inviscid flow. As physical model, we use a dispersive Boussinesq-type (BT) system for small-amplitude long waves with weak transverse variation. The problem is discretised in space using a wavelet-Galerkin method based on one-periodic Daubechies scaling functions. Assuming periodic boundary conditions, the discretised operators in spatial domain are circulant and skew-symmetric. These characteristics of discretised differential operators allow us to incorporate the Fast Fourier Transformation (FFT) in the matrix operations which results in a substantial improvement in the computational efficiency and accuracy of the numerical solver compared with the conventional finite difference or finite volume methods. We use a four-stage Runge–Kutta method to temporally discretise the governed spatially discretised differential equations. Several comparative test cases are conducted to validate the performance and efficiency of the proposed wavelet-Galerkin scheme for the BT model over flat beds relative to some existing analytical solutions and numerical results from a second-order finite difference method. We examine the numerical results of the BT system to investigate the two-way propagation of waves for some large L2-norm profiles of the initial free-surface elevation. We also assess the ability of the proposed method to predict the evolution and breaking of undular bores over a flat bed with a simple kinematic criterion. Moreover, we study (tsunami) wave runup and propagation by incorporating the effects of depth-varying topography in a simplified BT system in order to check the applicability of the approach to capture the interactions between the bathymetric features and the wet cells.

Numerical Scheme to Dispersive Shallow Water Waves

This paper studies dispersive shallow water waves modeled by Rosenau Korteweg-de Vries (KdV) Regularized long wave (RLW) equation or R-KdV-RLW equation that is considered with power law nonlinearity. The numerical algorithm is based on collocation finite element method with quintic B-splines. Test problems including the motion of solitary waves and shock waves are studied to validate the suggested method. Accuracy and efficiency of the proposed method are discussed by computing the numerical conserved laws and error norms L2 and L∞. A linear stability analysis based on a Fourier method shows that the numerical scheme is unconditionally stable.

Numerical simulation for treatment of dispersive shallow water waves with Rosenau-KdV equation

In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable.