Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations

In this paper, a numerical scheme based on the element-free Galerkin (EFG) method is proposed to find the numerical solutions of nonlinear sine-Gordon equation with Neumann boundary condition and generalized sinh-Gordon equation with Dirichlet boundary condition. In this scheme, a time stepping technique is used to approximate the time derivative terms of the given equations. Then, the penalty method is adopted to enforce the Dirichlet boundary condition and lastly, the EFG method is performed to establish the system of discrete equations. The convergence of the proposed scheme is derived theoretically and verified numerically by doing its error analysis. Numerical examples involving line and ring solitons are given to show the accuracy and efficiency of the scheme. The numerical results are in excellent agreement with the analytical solutions and/or previously reported numerical results.