The accuracy of particle approximation in Smoothed Particle Hydrodynamics (SPH) method decreases due to irregular particle distributions, especially for second-order derivatives. This study aims to enhance the accuracy of SPH method and analyze its convergence with irregular particle distributions. By establishing regularity conditions for particle distributions, we ensure that the local truncation error of traditional SPH formulations, including first and second derivatives, achieves second-order accuracy. Our proposed method, the volume reconstruction SPH method, guarantees these regularity conditions while preserving the discrete maximum principle. Benefiting from the discrete maximum principle, we conduct a rigorous global error analysis in the $L^\infty$-norm for the Poisson equation with variable coefficients, achieving second-order convergence. Numerical examples are presented to validate the theoretical findings.

翻译：光滑粒子流体动力学（SPH）方法中粒子近似的精度会因不规则粒子分布而降低，尤其对于二阶导数。本研究旨在提升SPH方法的精度并分析其在非规则粒子分布下的收敛性。通过建立粒子分布的规则性条件，我们确保传统SPH格式（包含一阶与二阶导数）的局部截断误差达到二阶精度。我们提出的体积重构SPH方法在保持离散极大值原理的同时，保证了这些规则性条件。借助离散极大值原理，我们对变系数泊松方程进行了$L^\infty$范数下的严格全局误差分析，实现了二阶收敛。数值算例验证了理论结果。