Numerical solution for series sine-Gordon equations using variational method and finite element approximation

This paper investigates numerical solutions for several kinds of sine-Gordon equations using variational method and finite element approximation. For the case of one-dimension and continuous time, a semi-discrete algorithm is proposed using Gauss-Legendre quadrature and Runge-Kutta method. Furthermore, the convergence of the algorithm is proved. Finally, several numerical examples are implemented and some simulation results are presented to show the efficiency of the scheme.