Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients

Abstract EN

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.