Smoothed particle hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the mathematics of SPH, a nonlocal relaxation to the conventional linear steady-state Stokes system for incompressible viscous flows. The nonlocal continuum model is characterized by a smoothing length \(\delta \) which measures the range of nonlocal interactions. It serves as a bridge between the discrete approximation schemes that involve a nonlocal integral relaxation and the local continuum models. We show that for a class of carefully chosen nonlocal operators, the resulting nonlocal Stokes equation is well-posed and recovers the original Stokes equation in the local limit when \(\delta \) approaches zero. For some other commonly used smooth kernels, there are risks in getting ill-posed continuum models that could lead to computational difficulties in practice. This leads us to discuss the implications of our finding on the design of numerical methods.

中文翻译：

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光滑粒子流体动力学数学：基于非局部斯托克斯方程的研究

平滑粒子流体动力学（SPH）是一种流行的数值技术，用于模拟复杂的流体流动。其关键要素之一是使用非局部积分松弛来进行局部微分。在连续水平上对相应的非局部模型进行数学分析可以提供对SPH的进一步理论理解。在有关SPH数学的一系列工作的这一部分中，我们介绍了不可压缩粘性流的传统线性稳态Stokes系统的非局部松弛。非局部连续模型的特征在于平滑长度\（\ delta \）用来衡量非本地互动的范围。它充当涉及非局部积分松弛的离散逼近方案与局部连续模型之间的桥梁。我们表明，对于一类精心选择的非局部算子，当\（\ delta \）接近零时，所得的非局部Stokes方程具有良好的位置并能在局部极限中恢复原始的Stokes方程。对于其他一些常用的平滑核，存在病态连续体模型的风险，这可能在实践中导致计算困难。这使我们讨论了我们的发现对数值方法设计的意义。