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arXiv 1608.05883physics.comp-phastro-ph.COastro-ph.GAphysics.flu-dyn
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光滑粒子流体力学一致性的新认识

A new insight into the consistency of smoothed particle hydrodynamics

Leonardo Di G. Sigalotti, Otto Rendón, Jaime Klapp, Carlos A. Vargas, Kilver Campos

备注：
27 pages. v2: Matches published version
发表信息：
Applied Mathematics and Computation 356 (2019) 50-73
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摘要：

本文解决了光滑粒子流体动力学（SPH）的一致性问题。一种新的误差分析是在$n$维空间中使用泊松求和公式，它使核和粒子近似误差的治疗相结合的方式。新的一致性积分关系推导出粒子近似对应的余弦傅立叶变换的经典已知的核近似的一致性条件。SPH插值参数，即平滑长度$h$和内核支持${\cal{N}}$内的粒子数上的误差界的函数依赖性被明确地证明，从中可以看出自然遵循的一致性条件。作为${\cal{N}}\to\infty$，粒子近似独立于$h$收敛到核近似，只要粒子质量与$h$成比例，作为$m\propto h^β$，与$β>n$成比例。这意味着作为$h\to 0$，联合极限$m\to 0$、${\cal{N}}\to\infty$和$N\to\infty$对于完全收敛到连续统是必要的，其中$N$是粒子的总数。分析还揭示了形式为$(\ln {\cal{N}})^{n}/{\cal{N}}$的主导误差项的存在，当${\cal{N}}\gg 1$时，该误差项渐进地倾向于$1/{\cal{N}}$，因为长期以来人们一直基于SPH和准蒙特卡罗估计之间的相似性进行推测。

英文摘要：

In this paper the problem of consistency of smoothed particle hydrodynamics (SPH) is solved. A novel error analysis is developed in $n$-dimensional space using the Poisson summation formula, which enables the treatment of the kernel and particle approximation errors in combined fashion. New consistency integral relations are derived for the particle approximation which correspond to the cosine Fourier transform of the classically known consistency conditions for the kernel approximation. The functional dependence of the error bounds on the SPH interpolation parameters, namely the smoothing length $h$ and the number of particles within the kernel support ${\cal{N}}$ is demonstrated explicitly from which consistency conditions are seen to follow naturally. As ${\cal{N}}\to\infty$, the particle approximation converges to the kernel approximation independently of $h$ provided that the particle mass scales with $h$ as $m\propto h^β$, with $β>n$. This implies that as $h\to 0$, the joint limit $m\to 0$, ${\cal{N}}\to\infty$, and $N\to\infty$ is necessary for complete convergence to the continuum, where $N$ is the total number of particles. The analysis also reveals the presence of a dominant error term of the form $(\ln {\cal{N}})^{n}/{\cal{N}}$, which tends asymptotically to $1/{\cal{N}}$ when ${\cal{N}}\gg 1$, as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates.

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