Implementation of meshless LBIE method to the 2D non-linear SG problem

In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. The method is based on the LBIE with moving least-squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time-stepping method is employed to deal with the time derivative and a simple predictor-corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non-linear problems in large domains. Copyright (C) 2009 John Wiley & Sons, Ltd.