Numerical accuracy and stability of semilinear Klein--Gordon equation in de Sitter spacetime

Numerical simulations of the semilinear Klein–Gordon equation in the de Sitter spacetime are performed. We use two structure-preserving discrete forms of the Klein–Gordon equation. The disparity between the two forms is the discretization of the differential term. We show that one of the forms has higher numerical stability and second-order numerical accuracy with respect to the grid, and we explain the reason for the instability of the other form.