Elsevier

Wave Motion

Volume 107, December 2021, 102805
Wave Motion

The local meshless collocation method for numerical simulation of shallow water waves based on generalized equal width (GEW) equation

https://doi.org/10.1016/j.wavemoti.2021.102805Get rights and content

Highlights

  • A high-order numerical solution has been developed for solving the generalized equal width equation.

  • The proposed algorithm is based on the meshless local collocation method. Also, a suitable basis functions is used to simulate of the mentioned PDE.

  • A fourth-order numerical procedure is used to solve the nonlinear system of ODEs.

  • For the first time this model is solved by a local meshless method.

  • The numerical results confirm that the proposed algorithm has high-accuracy.

Abstract

The current paper concerns to develop an efficient and robust numerical technique to solve the shallow water equation based on the generalized equal width (GEW) model. The considered model i.e. the generalized equal width (GEW) equation is a PDE that it can be classified in the category of hyperbolic PDEs. The solution of hyperbolic PDEs is similar to a fixed or moving wave. Thus, for solving these problems, a suitable numerical procedure that its basis functions are similar to a flat or shape wave should be selected. For this aim, the local collocation method via two different basis functions is utilized. First, the space derivative is approximated by the local collocation procedure that this manner yields a system of nonlinear ODEs depends on the time variable. Furthermore, the constructed system of ODEs is solved by a fourth-order algorithm to get high-numerical results. The mentioned process is applied on several test problems to verify the efficiency of the numerical formulation.

Introduction

In the current paper, we consider the following mathematical model ut+ɛupuxμuxxt=0,(x,t)Ω×(0,T],p1,with initial condition u(x,0)=u0(x),xΩ,and Ω=[a,b] and also the periodic boundary conditions u(a,t)=u(b,t),uxx(a,t)=uxx(b,t).

It is clear that most of real phenomena in mechanics, physics, chemistry etc are modeled by nonlinear PDEs. On the other hand, finding an analytical solution is an interested field for this category of applied problems but this aim may be not reached for all nonlinear PDEs  [1], [2]. Then, Benjamin [3] used it to explain long waves with small amplitudes on the surface of water in a channel. The delta-shaped basis functions-pseudospectral method is developed in [4] for solving nonlinear generalized equal width equation in shallow water waves. The RLW equation can be used to describe several phenomena such as plasma waves [5], shallow water waves [1]. The width (EW) equation is introduced by Morrison et al. [6] as an alternative for the RLW and well known KdV equations. The EW equation is investigated in [7], [8]. There are a few numerical procedures for GEW equation such as Petrov–Galerkin method [9], quadratic collocation method [10], cubic B-spline collocation method [11], cubic B-spline Galerkin approach [12] and another Petrov–Galerkin method [13]. Author of [14] developed tanh method based on the non-polynomial shape function for solving traveling wave solutions of nonlinear equations. Author of [15] employed a Legendre collocation method for solving distributed-order fractional optimal control problems. The main target of [16] is to develop interpolating element free Galerkin method for solving the Korteweg–de Vries–Rosenau-regularized long-wave equation. Authors of [17] developed a meshless method for solving the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. By using the improved moving least-square (IMLS) approximation, the improved element-free Galerkin (IEFG) method is investigated in [18] to solve diffusional drug release problems. The main aim of [19] is to combine the dimension splitting method with the improved element-free Galerkin (IEFG) method for simulating the 3D advection–diffusion problems. The complex variable reproducing kernel particle method (CVRKPM) is developed in [20] for solving two-dimensional variable coefficient advection–diffusion problems. The main aim of [21] is to propose the interpolating element-free Galerkin (IEFG) method for solving the three-dimensional (3D) elastoplasticity problems. An interpolating element-free Galerkin (IEFG) method is presented in [22] for solving the two-dimensional (2D) elastic large deformation problems. Also, authors of [23] developed the interpolating element-free Galerkin (IEFG) method for solving three-dimensional (3D) transient heat conduction problems. The main aim of [24] is to combine the dimension splitting method (DSM) with the improved complex variable element-free Galerkin (ICVEFG) method for solving three-dimensional (3D) elasticity equation. The improved complex variable element-free Galerkin (ICVEFG) method is proposed in [25] for solving the bending problem of thin plate on elastic foundations.

The tanh–coth method [26] is applied to obtain the soliton solutions of RLW equations, and thereafter, the approximation of finite domain interval is done by truncating the infinite domain interval. The multidimensional sine–Gordon (SG) equation in [27] has been simulated with various examples of linear and ring solitons with the development of numerical algorithms based on barycentric rational interpolation and local radial basis functions (RBFs). To solve the non-linear Schrödinger equation with constant and variable coefficients, the authors [28] first reduced the spatial derivatives using the local radial basis functions based on differential quadrature method (LRBF-DQM) and then solved the system obtained from ordinary nonlinear equations with the Runge–Kutta (RK-4) method. In [29] a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) has been used to solve the three-dimensional nonlinear wave equation. This method converts the three-dimensional nonlinear wave equation to an ordinary differential equation (ODE), which is solved by Runge–Kutta (SSP-RK54) method . The interested reader can see [30], [31], [32], [33], [34], [35], [36], [37], [38] to get more information for nonlinear waves.

Section snippets

Local collocation idea based on RBFs and moving Kriging approximation

In the current section, we want to explain the pseudo-spectral method based upon the RBFs and B-spline functions.

Full-discrete scheme

To obtain the full-discrete scheme, the space derivative is approximated by the local meshless method so a semi-discrete scheme, i.e. system of ODEs related to the time derivative, is constructed. Then, the achieved system of ODEs is solved by using the ETDRK4 algorithm [51].

To derive the full-discrete scheme of the following problem ut+ɛupuxμuxxt=0,(x,t)Ω×(0,T],p1,we change it to Iμ2x2u(x,t)t=ɛup(x,t)u(x,t)x.Let for every distributed node xs, there be number of ns nodes in its

Numerical analysis

We check the stability, ability and convergence rate of the recent numerical plane for several examples based upon the L error. Also, we check the conservation of three qualities:

    Conservation of Mass:

    I1=abu(x,T)dxhi=1Nu(xi,T).

    Conservation of Momentum:

    I2=abu2(x,T)+μux2(x,T)dxhi=1Nu2(xi,T)+μux2(xi,T).

    Conservation of Energy:

    I3=abup+2(x,T)dxhi=1Nup+2(xi,T).

Conclusion

In this paper, we developed a new meshless numerical procedure for solving the shallow water equation based on the generalized equal width (GEW) model. The solution of the mentioned model is sharp in a center point and it approaches to the zero for other points of the computational domain. The space derivative is discretized by two different basis functions based upon the local collocation idea. In the proposed numerical procedure, we do not have any numerical integration and we just need to

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We would like to express our very great appreciation to both reviewers for their valuable and constructive suggestions to improve the quality of the paper.

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    The contribution of all authors in the preparation of this article is the same.

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