  The computational analysis of the Cauchy problem for semi-linear Klein-Gordon
equations in the de Sitter spacetime is considered. Several simulations are
performed to show the time-global behaviors of the solutions of the equations
in the spacetime based on the structure-preserving scheme. It is remarked that
the sufficiently large Hubble constant yields the strong diffusion-effect which
gives the long and stable simulations for the defocusing semi-linear terms. The
reliability of the simulations is confirmed by the preservation of the
numerically modified Hamiltonian of the equations.
