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

Abstract

• A meshless method is developed for some generalized Klein–Gordon equations. • A convergent iterative scheme is performed to tackle the nonlinearity. • Examples involving five kinds of nonlinear equations and solitons are given. • Numerical results indicate that the method is about second-order convergence in both time and space. 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.