Numerical Solution of Sine-Gordon Equation with the Local Kriging Meshless Method

Pengfei Guo,Ariunkhishig Boldbaatar,Pengxiang Dai,Dutao Yi

doi:10.1155/2020/9057387

Pengfei Guo, Ariunkhishig Boldbaatar + Show 2 more

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https://doi.org/10.1155/2020/9057387

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Abstract

This paper develops a local Kriging meshless solution to the nonlinear 2 + 1-dimensional sine-Gordon equation. The meshless shape function is constructed by Kriging interpolation method to have Kronecker delta function property for the two-dimensional field function, which leads to convenient implementation of imposing essential boundary conditions. Based on the local Petrov–Galerkin formulation and the center difference method for time discretization, a system of nonlinear discrete equations is obtained. The numerical examples are presented and the numerical solutions are found to be in good agreement with the results in the literature to validate the ability of the present meshless method to handle the 2 + 1-dimensional sine-Gordon equation related problems.

Highlights

The Local Kriging Meshless Formulation for the 2D Sine-Gordon EquationWe set c0 to 1 because its value has little influence on the shape functions, whereas a0 is evaluated according to the influence domain size (ds) linked to nodes a0 α0 · ds, α0 ≥ 1􏼁,
Introduction e nonlinear sineGordon equation (SGE), a type of hyperbolic partial differential equation, is often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, such as nonlinear waves, propagation of fluxons and dislocation of metals [1,2,3,4]
We present the local Kriging meshless method [44,45,46,47] to seek the numerical solutions of the 2 + 1dimensional nonlinear sine-Gordon equation. e present meshless approach is based on the Kriging interpolation technique and the local Petrov–Galerkin formulation, which leads to the shape functions having the Kronecker delta property and to a convenient implementation required for imposing the essential boundary conditions, like in the traditional finite element analysis

Summary

The Local Kriging Meshless Formulation for the 2D Sine-Gordon Equation

We set c0 to 1 because its value has little influence on the shape functions, whereas a0 is evaluated according to the influence domain size (ds) linked to nodes a0 α0 · ds, α0 ≥ 1􏼁,. We apply the local Petrov–Galerkin formulation to construct the weak form of the governing equation over the pre-established local subdomains (Ωq) associated with the nodes in the global problem domain Ω (see Figure 1). According to the Kriging interpolation method and the constructed shape function (equation (5)) presented, the field function u(x) to be determined can be approximated in terms of nodal values at the n nodes that are included in the support domain associated with the point at x as follows:. By conducting the above iterative procedure to solve the equation (equation (28)) till the desired time level, we achieve the numerical solutions to the 2 + 1-dimensional nonlinear sine-Gordon equation

Example 1

Example 2

Concluding Remarks