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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.

The nonlinear sine-Gordon 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 [

As one of the crucial equations in nonlinear science, the sine-Gordon equation has been constantly investigated and solved numerically and analytically in recent years [

During recent decades, meshless methods have also been developed for solving the nonlinear sine-Gordon equations, which include the mesh-free

In this study, we present the local Kriging meshless method [

In the present study, we seek to acquire a meshless approximation to the damped 2 + 1-dimensional sine-Gordon equation [

The boundary conditions linked to the sine-Gordon equation (equation (

At time

The problem domain

The matrices

In terms of the basis function at nodes in the support domain, the matrix

The matrix

In this numerical investigation, we adopt the Gauss model to construct the shape functions via the Kriging interpolation scheme [

In equation (

In the problem domain

In the present study, we apply the local Petrov–Galerkin formulation to construct the weak form of the governing equation over the pre-established local subdomains

Local quadrature, support, and test function domains.

By decomposing the subdomain boundary

According to the Kriging interpolation method and the constructed shape function (equation (

The time and space derivatives of the filed function are expressed in the following form of

Substituting the approximate field function

In order to construct the global system equation, the assembling process similar to the finite difference method has been used for all

Apparently, equation (

In the present study, the time derivatives of the global system equations (equation (

The above equation can be rewritten as

When

We can set

By conducting the above iterative procedure to solve the equation (equation (

In this section, we apply the local Kriging meshless method to several numerical examples regarding two-dimensional line and ring solitons.

In the case of an undamped 2D sine-Gordon equation

Superposition of two orthogonal line solitons (

Contour for superposition of two orthogonal line solitons (

In order to investigate the effects of the dissipative term on the behavior, the same numerical example is used with

Contour for superposition of two orthogonal line solitons (

In order to quantitatively compare the numerical results with the literature, we adopt the energy for an undamped sine-Gordon equation

The integration of the above equation over the problem domain

The energy

– | 175.3027 | 175.3107 | 175.3289 | 175.3743 | 175.3921 | 175.4131 | 175.7865 | |

175.5745 | 175.2927 | 175.2761 | 175.3188 | 175.3637 | 175.3899 | 175.4045 | 175.6831 | |

175.7503 | 175.7515 | 175.8155 | 175.9262 | 176.0511 | 176.2110 | 176.3888 | 176.5849 |

The convergence analysis of the present method is conducted based on the energy,

Relationship between the energy

For the numerical case, in which

Circular ring solitons in terms of

Contours of circular ring solitons in terms of

The perturbation of a line soliton is studied in the numerical example with

Symmetric perturbation of static line soliton in terms of

In this study, the local Kriging meshless method has been extended to the 2 + 1-dimensional nonlinear sine-Gordon equation. The Kriging interpolation technique is used to approximate the two-dimensional field function, which leads to convenient implementation of imposing essential boundary conditions. A system of nonlinear discrete equations can be established based on the adoption of the local Petrov–Galerkin formulation and the center difference method for time discretization. The nonlinear algebraic equations are solved by applying the iterative technique and a predictor-corrector scheme. The numerical results are in good agreement with those available in the literature. The present meshless method is thus a potential alternative to other numerical methods, such as the finite element method and finite difference method, for dealing with a 2 + 1-dimensional nonlinear sine-Gordon equation.

All results have been obtained by conducting the numerical procedure and the ideas can be shared for the researchers.

The authors declare that they have no conflicts of interest.

This work was supported by Hainan Provincial Natural Science Foundation of China (Grant no. 119MS039).