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.

中文翻译：

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广义粒子法中近似算子的截断误差估计

为了便于粒子方法的数值分析，我们导出了广义粒子方法中近似算子的截断误差估计。在这里，广义粒子方法被定义为无网格数值方法，它通常包括其他常规粒子方法，例如平滑粒子流体动力学或移动粒子半隐式方法。通过基于域的 Voronoi 分解的两个新指标以及参考权重函数的两个假设，提出了离散参数的新规律。然后，在广义粒子方法中为插值、近似梯度算子和近似拉普拉斯算子导出截断误差估计。这些估计的收敛速度是根据它们出现在规律性和假设中的频率来确定的。最后，对估计值进行了数值计算，结果表明与理论结果非常吻合。