# Truncation error estimates of approximate operators in a generalized particle method

## TL;DR

This paper derives and verifies truncation error estimates for approximate operators in a generalized particle method, enhancing the understanding of its numerical accuracy and convergence behavior.

## Contribution

It introduces a new regularity framework and error estimates for operators in a generalized particle method, unifying analysis across various particle techniques.

## Key findings

- Error estimates align with theoretical convergence rates.
- Numerical results confirm the accuracy of the derived estimates.
- The framework applies to multiple particle methods, including SPH.

## Abstract

To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method that typically includes other conventional particle methods, such as smoothed particle hydrodynamics or moving particle semi-implicit methods. A new regularity of discrete parameters is proposed via two new indicators based on the Voronoi decomposition of the domain along with two hypotheses of reference weight functions. Then, truncation error estimates are derived for an interpolant, approximate gradient operator, and approximate Laplace operator in the generalized particle method. The convergence rates for these estimates are determined based on the frequency with which they appear in the regularity and hypotheses. Finally, the estimates are computed numerically and the results are shown to be in good agreement with the theoretical results.