The local meshless collocation method for numerical simulation of shallow water waves based on generalized equal width (GEW) equation
Introduction
In the current paper, we consider the following mathematical model with initial condition and and also the periodic boundary conditions
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 we change it to Let for every distributed node , there be number of nodes in its
Numerical analysis
We check the stability, ability and convergence rate of the recent numerical plane for several examples based upon the error. Also, we check the conservation of three qualities:
- Conservation of Mass:
- Conservation of Momentum:
- Conservation of Energy:
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.