An implicit Lie-group iterative scheme for solving the nonlinear Klein–Gordon and sine-Gordon equations

In this article, the nonlinear Klein–Gordon and sine-Gordon equations are solved by pondering the semi-discretization numerical schemes and then, the resulting ordinary differential equations at the discretized spaces are numerically integrated toward the time direction by using the implicit Lie-group iterative method to find the unknown physical quantity. When six numerical experiments are examined, we reveal that the present implicit Lie-group iterative scheme is applicable to the nonlinear Klein–Gordon and sine-Gordon equations and convergent very fast at each time marching step, and the accuracy is raised several orders, of which the numerical results are rather accurate, effective and stable.