Klein-Gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton high-order absorbing boundary conditions. The space discretization is reached by means of finite differences on a uniform grid, with fourth order inside the computational domain. Time integration is made by means of exponential splitting schemes that are efficient and easy to implement. Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided. The influence of the dispersion coefficient on the error and the behavior of the absorption error are also studied.