# On the numerical experiments of the Cauchy problem for semi-linear Klein-Gordon equations in the de Sitter spacetime

## TL;DR

This paper presents numerical experiments analyzing the long-term behavior of solutions to semi-linear Klein-Gordon equations in de Sitter spacetime, highlighting the effects of the Hubble constant and the importance of structure-preserving schemes.

## Contribution

It introduces a structure-preserving numerical scheme for simulating the equations and demonstrates its effectiveness in capturing long-term dynamics and diffusion effects.

## Key findings

- Large Hubble constant induces strong diffusion effects.
- The scheme preserves the numerically modified Hamiltonian.
- Simulations show stable, long-term behavior of solutions.

## Abstract

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.