Optimal error estimates of fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation in the nonrelativistic regime

Two fourth-order compact finite difference schemes including a Crank-Nicolson one and a semi-implicit one are derived for solving the nonlinear Klein-Gordon equations in the nonrelativistic regime. The optimal error estimates and the strategy in choosing time step are rigorously analyzed, and the energy conservation in the discrete sense is also studied. Under proper assumption on the analytical solutions, the errors of the two schemes both are proved to be of O(h(4)+tau(2)/epsilon(6)) with mesh size h and time-step tau. Numerical simulations are provided to confirm the theoretical analysis.