  We perform some simulations of the semilinear Klein--Gordon equation in the
de Sitter spacetime. We reported the accurate numerical results of the equation
with the structure-preserving scheme (SPS) in an earlier publication (Tsuchiya
and Nakamura in J. Comput. Appl. Math. \textbf{361}: 396--412, 2019). To
investigate the factors for the stability and accuracy of the numerical results
with SPS, we perform some simulations with three discretized formulations. The
first formulation is the discretized equations with SPS, the second one is with
SPS that replaces the second-order difference as the standard second-order
central difference, and the third one is with SPS that replaces the discretized
nonlinear term as the standard discretized expression. As a result, the above
two replacements in SPS are found to be effective for accurate simulations. On
the other hand, the ingenuity of replacing the second-order difference in the
first formulation is not effective for maintaining the stability of the
simulations.
