# Quantitative evaluations of stability and convergence for solutions of semilinear Klein--Gordon equation

## TL;DR

This paper conducts simulations of the semilinear Klein--Gordon equation to quantitatively evaluate the stability and convergence of numerical solutions, analyzing thresholds based on initial amplitude and mass.

## Contribution

It introduces specific quantitative evaluation methods for stability and convergence and identifies optimal thresholds by varying initial conditions.

## Key findings

- Proposed new methods for stability and convergence evaluation.
- Identified thresholds for initial amplitude and mass affecting solution stability.
- Validated evaluation methods through numerical simulations.

## Abstract

We perform some simulations of the semilinear Klein--Gordon equation with a power-law nonlinear term and propose each of the quantitative evaluation methods for the stability and convergence of numerical solutions. We also investigate each of the thresholds in the methods by varying the amplitude of the initial value and the mass, and propose appropriate values.