A new insight into the consistency of the SPH interpolation formula

Leonardo Di G. Sigalotti, Otto Rendón, Jaime Klapp, Carlos A. Vargas and Fidel Cruz

Applied Mathematics and Computation, 2019, vol. 356, issue C, 50-73

Abstract: In this paper, the consistency of the smoothed particle hydrodynamics (SPH) interpolation formula is investigated by analytical means. A novel error analysis is developed in n-dimensional space using the Poisson summation formula, which enables the simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions. New consistency integral relations are derived for the particle approximation, which correspond to the cosine Fourier transform of the kernel consistency conditions. 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, N, is demonstrated explicitly from which consistency conditions arise. As N→∞, the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m∝hβ with β > n, where n is the spatial dimension. This implies that as h → 0, the joint limit m → 0, N→∞, and N → ∞ is necessary for complete convergence to the continuum, where N is the total number of particles. The analysis also reveals a dominant error term of the form (lnN)n/N for finite N, as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates. When N≫1, the error of the SPH interpolant decays as N−1 independently of the dimension. This ensures approximate partition of unity of the kernel volume.

Keywords: Particle methods; Numerical integration; Error analysis and interval analysis; Fourier transforms; Error bounds; Stability and convergence of numerical methods (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S009630031930219X
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:356:y:2019:i:c:p:50-73

DOI: 10.1016/j.amc.2019.03.018

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().