Meshless numerical analysis of a class of nonlinear generalized Klein-Gordon equations with a well-posed moving least squares approximation

This paper presents a meshless method for the numerical solution of a class of nonlinear generalized Klein-Gordon equations. In this method, a time discrete technique is first adopted to discretize the time derivatives, and then a well-posed moving least squares (WP-MLS) approximation using shifted and scaled orthogonal basis functions is developed to approximate the spatial derivatives. To deal with the nonlinearity, an iterative scheme is presented and the corresponding convergence is discussed theoretically. Numerical examples involving Klein-Gordon, Dodd-Bullough-Mikhailov, sine-Gordon, double sine-Gordon and sinh-Gordon equations, and line and ring solitons are provided to illustrate the performance and efficiency of the method.