Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

Abstract

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor–corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.