Nonlinear Shallow Water Wave Equations Research Articles

Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Auto-Bäcklund Transformation with the Solitons and Similarity Reductions for a Generalized Nonlinear Shallow Water Wave Equation

Local well-posedness and blow-up criterion to a nonlinear shallow water wave equation

<abstract><p>The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.</p></abstract>

The dynamical property of a nonlinear shallow water wave equation with inhomogeneous boundary conditions

A nonlinear shallow water wave equation containing the Fornberg–Whitham model is considered. The phase portrait analytical technique is employed to establish the existence of the smooth, peaked and cusped solitary wave solutions of the equation under inhomogeneous boundary conditions. Asymptotic and numerical analysis illustrates the dynamical features for the smooth, peaked and cusped solitary wave solutions. Our results are helpful to further understand the dynamical tendency of the solutions when the space variable tends to positive or negative infinite.

Approximate solutions to shallow water wave equations by the homotopy perturbation method coupled with Mohand transform

In this paper, the Mohand transform-based homotopy perturbation method is proposed to solve two-dimensional linear and non-linear shallow water wave equations. This approach has been proved suitable for a broad variety of non-linear differential equations in science and engineering. The variation trend of the water surface elevation at different time levels and depths are given by some graphs. Moreover, the obtained solutions are compared with the existing results, which show higher efficiency and fewer computations than other approaches studied in the literature.

The entropy weak solution to a nonlinear shallow water wave equation including the Degasperis-Procesi model

<abstract><p>A nonlinear model, which characterizes motions of shallow water waves and includes the famous Degasperis-Procesi equation, is considered. The essential step is the derivation of the $ L^2(\mathbb{R}) $ uniform bound of solutions for the nonlinear model if its initial value belongs to space $ L^2(\mathbb{R}) $. Utilizing the bounded property leads to several estimates about its solutions. The viscous approximation technique is employed to establish the well-posedness of entropy weak solutions.</p></abstract>

Quasi-Linear Model of Tsunami Run-Up on a Beach with a Seafloor Described by the Piecewise Continuous Function

The purpose of this paper is to propose the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. We begin with the one-dimensional nonlinear shallow-water wave equations, which we consider over a beach of complex geometry that can be modeled by a piecewise continuous function, along with several natural initial and boundary conditions. The primary obstacle in solving this problem is the moving boundary associated with the shoreline motion. To avoid this difficulty, we replace the moving boundary with a stationary boundary by applying a transformation to the spatial variable of the computational domain. A characteristic feature of any tsunami problem is the smallness of the parameter ε=η0/h0, where η0 is the characteristic amplitude of the wave, and h0 is the characteristic depth of the ocean. The presence of this small parameter enables us to effectively linearize the problem by using the method of perturbations, which leads to an analytical solution via an integral transformation. This analytical solution assumes that there is no wave breaking. In light of this assumption, we introduce the wave no-breaking criterion and determine bounds for the applicability of our theory. The proposed model can be readily used to investigate the tsunami run-up and draw-down for different sea bottom profiles. The novel particular solution, when the seafloor is described by the piecewise linear function, is obtained, and the effects of the different beach profiles and initial wave locations are considered.

Symmetric waves are traveling waves of some shallow water scalar equations

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis‐Procesi equation. Journal of Differential Equations. 2020;269(10):7730‐7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr (A new highly nonlinear shallow water wave equation. Journal of Evolution Equations. 2016;16(3):539‐556) are indeed traveling waves, with further information on their steady structures. We also provide a simple proof that symmetric waves are traveling waves to the free surface evolution equation of moderate amplitude waves in shallow water.

A comparative study about the propagation of water waves with fractional operators

In the current research article, the truncated M-fractional derivative and Atangana–Baleanu derivative in Riemann–Liouville sense is practiced in the fractional modeling of the weakly non-linear shallow water wave equation. This paper provides some traveling wave transformations for the conversion of the partial fractional differential equation into an ordinary differential equation concerning fractional operators. In order to portray the wave propagation in weakly non-linear and dispersive media, the weakly non-linear shallow water wave model has been utilized. The proficient approach new extended direct algebraic scheme is used to find the solitonic structures. It is observed that the acquired solutions are purely new and have full novelty. The obtained results are carrying dark, dark singular, bright singular, and bright-dark singular solutions, which are obtained via Mathematica. The graphical comparison is also depicted in the perspective of operating fractional differential definitions with the selection of appropriate parametric values.

Dynamical control on the Adomian decomposition method for solving shallow water wave equation

The aim of this study is to apply a novel technique to control the accuracy and error of the Adomian decomposition method (ADM) for solving nonlinear shallow water wave equation. The ADM is among semi-analytical and powerful methods for solving many mathematical and engineering problems. We apply the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method which is based on stochastic arithmetic (SA). Also instead of applying mathematical packages we use the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. In this library we will write all codes using C++ programming codes. Applying the method we can find the optimal numerical results, error and step of the ADM and they are the main novelties of this research. The numerical results show the accuracy and efficiency of the novel scheme.

Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation

In this paper, the nonlinear shallow water wave equation is illustrated. The famous semi-analytical method, homotopy analysis method (HAM) is applied for solving this equation. The main novelty, of this study is to validate the numerical results using the stochastic arithmetic, the CESTAC method and the CADNA library. Based on this method, we can find the optimal iteration of the HAM, optimal approximation of the shallow water wave equation and optimal error. The main theorem of the CESTAC method is proved. Based on this theorem, we can show that the number of common significant digits for two successive approximations are almost equal to the number of common significant digits for exact and approximate solutions. Thus instead of traditional absolute error to show the accuracy of method we can apply the new termination criterion depends on two successive approximations. In order to find the convergence region of the HAM, several ħ-curves are demonstrated.

Invariant Subspace Classification and Exact Explicit Solutions to a Class of Nonlinear Wave Equation

In this paper, the invariant subspace classification of a class of nonlinear shallow water wave equation is given, then some exact explicit solutions to the nonlinear equation are provided by using the invariant subspace method. This method is a dynamical system method by nature, for its key step is to transform a nonlinear partial differential equation (PDE) into ordinary differential equation (ODE) systems, then by solving the ODE systems, the exact solutions to the nonlinear PDE are obtained.

Homotopy perturbation method for predicting tsunami wave propagation with crisp and uncertain parameters

PurposeThe purpose of this paper is to find the solution of classical nonlinear shallow-water wave (SWW) equations in particular to the tsunami wave propagation in crisp and interval environment.Design/methodology/approachHomotopy perturbation method (HPM) has been used for handling crisp and uncertain differential equations governing SWW equations.FindingsThe wave height and depth-averaged velocity of a tsunami wave in crisp and interval cases have been obtained.Originality/valuePresent results by HPM are compared with the existing solution (in crisp case), and they are found to be in good agreement. Also, the residual error of the solutions is found for the convergence conformation and reliability of the present results.

Symmetry Reductions, Dynamical Behavior and Exact Explicit Solutions to a Class of Nonlinear Shallow Water Wave Equation

By using Lie symmetry analysis and dynamical systems method for a class of nonlinear shallow water wave equation, the exact solutions based on the Lie group method are provided. Especially, the bifurcations and exact explicit parametric representations of the traveling solutions are given, and the possible solitary wave solutions and many uncountable infinite periodic wave solutions to the nonlinear equation are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, we give some exact analytic solutions by using the power series method. This result enriches the types of solutions of nonlinear shallow water wave equation and has important physical significance for further study of this kind of equation.

A class of exact nonlinear traveling wave solutions for shallow water with a non-stationary bottom surface

Abstract The GN nonlinear shallow water wave equations developed by Green and Naghdi (1976) [5,6] are valid for a non-stationary, non-uniform bottom surface and a non-uniform pressure on the top surface. In contrast, the S nonlinear shallow water wave equations developed by Serre (1953) [12,13] for uniform depth and later generalized by Seabra-Santos et al. (1987) for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface. This paper develops a class of exact nonlinear traveling wave solutions of the GN equations for a non-stationary, non-uniform bottom surface. Also, the explicit expressions for the pressure acting on the bottom surface and the average pressure through the depth in the GN equations are used to place physical restrictions on the motion which ensure that these pressures remain non-negative preventing cavitation.

Solution of interval shallow water wave equations using homotopy perturbation method

PurposeThis paper aims to present solutions of uncertain linear and non-linear shallow water wave equations. The uncertainty has been taken as interval and one-dimensional interval shallow water wave equations have been solved by homotopy perturbation method (HPM). In this study, basin depth and initial conditions have been taken as interval and the single parametric concept has been used to handle the interval uncertainty.Design/methodology/approachHPM has been used to solve interval shallow water wave equation with the help of single parametric concept.FindingsPreviously, few authors found solution of shallow water wave equations with crisp basin depth and initial conditions. But, in actual sense, the basin depth, as well as initial conditions, may not be found in crisp form. As such, here these are considered as uncertain in term of intervals. Hence, interval linear and non-linear shallow water wave equations are solved in this study using single parametric concept-based HPM.Originality/valueAs mentioned above, uncertainty is must in the above-titled problems due to the various parametrics involved in the governing differential equations. These uncertain parametric values may be considered as interval. To the best of the authors’ knowledge, no work has been reported on the solution of uncertain shallow water wave equations. But when the interval uncertainty is involved in the above differential equation, then direct methods are not available. Accordingly, single parametric concept-based HPM has been applied in this study to handle the said problems.

Communication-aware adaptive Parareal with application to a nonlinear hyperbolic system of partial differential equations

In the strong scaling limit, the performance of conventional spatial domain decomposition techniques for the parallel solution of PDEs saturates. When sub-domains become small, halo-communication and other overhead come to dominate. A potential path beyond this scaling limit is to introduce domain-decomposition in time, with one such popular approach being the Parareal algorithm which has received a lot of attention due to its generality and potential scalability. Low efficiency, particularly on convection dominated problems, has however limited the adoption of the method. In this paper we demonstrate trough large-scale numerical experiments that it is possible not only to obtain time-parallel speedup on the non-linear shallow water wave equation, but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. Two factors were essential in achieving this. First, for Parareal to converge on the hyperbolic problem we used an approximate Riemann solver as the preconditioner. This preconditioner introduces only dissipative errors with respect to the 3rd order accurate WENO-RK discretization used to solve the PDE system. The preconditoner is comparatively expensive and convergence is slow unless time-subdomains are short. We therefore introduce a new scheduler that we denote Communication Aware Adaptive Parareal (CAAP). CAAP increases obtainable speed-up by minimizing the time-subdomain length without making communication of time-subdomains too costly whilst also adaptively overlapping consecutive cycles of Parareal so to mitigate the impact of a relatively expensive coarse operator.

Maximum run-up behavior of tsunamis under non-zero initial velocity condition

The tsunami run-up problem is solved non-linearly under the most general initial conditions, that is, for realistic initial waveforms such as N-waves, as well as standard initial waveforms such as solitary waves, in the presence of initial velocity. An initial-boundary value problem governed by the non-linear shallow-water wave equations is solved analytically utilizing the classical separation of variables technique, which proved to be not only fast but also accurate analytical approach for this type of problems. The results provide important information on maximum tsunami run-up qualitatively. We observed that, although the calculated maximum run-ups increase significantly, going as high as double that of the zero-velocity case, initial waves having non-zero fluid velocity exhibit the same run-up behavior as waves without initial velocity, for all wave types considered in this study.

Development of a Tsunami Inundation Analysis Model for Urban Areas Using a Porous Body Model

To evaluate tsunami hazards with strong locality in urban areas, this study developed a novel tsunami inundation model based on nonlinear shallow water wave equations and a porous body model (PBM). By applying a kinematic boundary condition that includes both porosity and surface permeability of the porous medium, the proposed model could accurately incorporate geometric effects such as the flow anisotropy caused by the distributions of buildings. The proposed PBM demonstrated as good accuracy for the inundation heights around buildings near the coastline as with a conventional three-dimensional simulation with high resolution. In addition, the model showed its capability to reproduce a tsunami’s essential behaviors in urban areas. In particular, the amplification effect of flow velocity along straight roads surrounded by buildings was reasonably reproduced. It can be expected that the present model can become a useful tool to accurately evaluate the tsunami risks in urban areas.

Comparison of solutions of linear and non-linear shallow water wave equations using homotopy perturbation method

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

New Analytical Solution for Nonlinear Shallow Water-Wave Equations

We solve the nonlinear shallow water-wave equations over a linearly sloping beach as an initial-boundary value problem under general initial conditions, i.e., an initial wave profile with and without initial velocity. The methodology presented here is extremely simple and allows a solution in terms of eigenfunction expansion, avoiding integral transform techniques, which sometimes result in singular integrals. We estimate parameters, such as the temporal variations of the shoreline position and the depth-averaged velocity, compare with existing solutions, and observe perfect agreement with substantially less computational effort.