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

TL;DR

This paper compares two structure-preserving numerical schemes for the semilinear Klein--Gordon equation in de Sitter spacetime, identifying which scheme offers better stability and accuracy for simulations.

Contribution

It introduces and analyzes two discretization methods, demonstrating that one achieves higher stability and second-order accuracy, explaining the causes of instability in the other.

Findings

One discretization form has higher numerical stability.

The stable form achieves second-order accuracy.

The instability of the other form is explained.

Abstract

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.