Approximate solution of nonlinear klein-gordon equation using sobolev gradients

The nonlinear klein-gordon equation (kge) models many nonlinear phenomena. in this paper,we propose a scheme for numerical approximation of solutions of the one-dimensional nonlinear kge. a common approach to find a solution of a nonlinear system is to first linearize the equations by successive substitution or the newton iteration method and then solve a linear least squares problem. here,we show that it can be advantageous to form a sum of squared residuals of the nonlinear problem and then find a zero of the gradient. our scheme is based on the sobolev gradient method for solving a nonlinear least square problem directly. the numerical results are compared with lattice boltzmann method (lbm). the l2,l∞,and root-mean-square (rms) values indicate better accuracy of the proposed method with less computational effort. © 2016 nauman raza et al.