Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients. (2nd March 2016) - Raza, Nauman; Butt, Asma Rashid; Javid, Ahmad
Abstract : 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 L 2, L ∞, and Root-Mean-Square (RMS) values indicate better accuracy of the proposed method with less computational effort.