SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1239-1262, January 2021.
The smoothed particle hydrodynamics (SPH) method is a popular, kernel-based discretization method for fluid-flow problems. Despite its frequent use, mathematical understanding is still limited. In [T. Franz and H. Wendland, SIAM J. Math. Anal., 50 (2018), pp. 4752--4784] we proved convergence for a specific flow problem under appropriate conditions on the underlying kernel. The kernel has to satisfy so-called moment and approximation conditions. We also showed that the generalized Wendland kernels satisfy these conditions in odd space dimensions. In this paper, we will significantly improve the above results in the following ways. We will show that the results also hold in even space dimensions. We will show that the generalized Wendland kernels satisfy the approximation condition of any order, which means that for these kernels we can eliminate the dependence of the convergence rate on the approximation condition. We will show that the standard Wendland kernels, though they perform numerically similarly, do not satisfy the approximation condition.

中文翻译：

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光滑粒子流体动力学方法收敛性的改进结果

SIAM数学分析杂志，第53卷，第2期，第1239-1262页，2021年1月。
平滑粒子流体动力学（SPH）方法是一种流行的基于核的离散化方法，用于解决流体流动问题。尽管经常使用，但是数学理解仍然有限。在[T. Franz和H.Wendland，SIAM J. Math。Anal。，50（2018），pp.4752--4784]中，我们证明了在适当条件下底层内核上特定流问题的收敛性。内核必须满足所谓的矩和逼近条件。我们还表明，广义的Wendland核在奇数空间维上满足这些条件。在本文中，我们将通过以下方式显着改善上述结果。我们将证明结果也适用于均匀的空间尺寸。我们将证明广义的Wendland核满足任何阶的逼近条件，这意味着对于这些内核，我们可以消除收敛速度对近似条件的依赖。我们将证明标准Wendland内核尽管在数值上表现相似，但不满足近似条件。