Numerical simulations of semilinear Klein--Gordon equation in the de Sitter spacetime with structure-preserving scheme

TL;DR

This paper investigates the stability and accuracy of structure-preserving numerical schemes for simulating the semilinear Klein-Gordon equation in de Sitter spacetime, comparing different discretization approaches.

Contribution

It evaluates the effects of various discretization modifications within the structure-preserving scheme on simulation stability and accuracy.

Findings

Replacing the nonlinear term with standard discretization improves accuracy.

Replacing the second-order difference does not enhance stability.

The original structure-preserving scheme maintains stability better than modified versions.

Abstract

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.