A truncation error analysis has been developed for the approximation of spatial derivatives in one-dimensional Smoothed Particle Hydrodynamics (SPH). For this purpose, the SPH interpolation is understood as a two-step process of smoothing and discretisation. As smoothing length is reduced while maintaining a constant ratio of particle spacing Δx to smoothing length h, error decays as h2; however, there is a finite limiting discretisation error. If particle spacing Δx is reduced while holding a constant h, error decreases at a rate which depends on the kernel function's smoothness at its boundaries. When particles are distributed non-uniformly, error behaviour is complex, and discretisation error can diverge as h is reduced. A first-order consistent SPH method is shown to remove this behaviour. The findings of the theoretical analysis are confirmed by numerical experiments.
%0 Conference Paper
%1 quinlan2005analysis
%A Quinlan, Nathan
%A Basa, Mihai
%A Lastiwka, Martin
%B 17th AIAA Computational Fluid Dynamics Conference
%D 2005
%I AIAA
%K 76m28-particle-methods-and-lattice-gas-methods-in-fluid-mechanics smoothed-particle-hydrodynamics
%N 4622
%R 10.2514/6.2005-4622
%T An Analysis of Accuracy in One-Dimensional Smoothed Particle Hydrodynamics
%U https://arc.aiaa.org/doi/10.2514/6.2005-4622
%X A truncation error analysis has been developed for the approximation of spatial derivatives in one-dimensional Smoothed Particle Hydrodynamics (SPH). For this purpose, the SPH interpolation is understood as a two-step process of smoothing and discretisation. As smoothing length is reduced while maintaining a constant ratio of particle spacing Δx to smoothing length h, error decays as h2; however, there is a finite limiting discretisation error. If particle spacing Δx is reduced while holding a constant h, error decreases at a rate which depends on the kernel function's smoothness at its boundaries. When particles are distributed non-uniformly, error behaviour is complex, and discretisation error can diverge as h is reduced. A first-order consistent SPH method is shown to remove this behaviour. The findings of the theoretical analysis are confirmed by numerical experiments.
@inproceedings{quinlan2005analysis,
abstract = {A truncation error analysis has been developed for the approximation of spatial derivatives in one-dimensional Smoothed Particle Hydrodynamics (SPH). For this purpose, the SPH interpolation is understood as a two-step process of smoothing and discretisation. As smoothing length is reduced while maintaining a constant ratio of particle spacing Δx to smoothing length h, error decays as h2; however, there is a finite limiting discretisation error. If particle spacing Δx is reduced while holding a constant h, error decreases at a rate which depends on the kernel function's smoothness at its boundaries. When particles are distributed non-uniformly, error behaviour is complex, and discretisation error can diverge as h is reduced. A first-order consistent SPH method is shown to remove this behaviour. The findings of the theoretical analysis are confirmed by numerical experiments.},
added-at = {2023-02-16T02:41:40.000+0100},
author = {Quinlan, Nathan and Basa, Mihai and Lastiwka, Martin},
biburl = {https://www.bibsonomy.org/bibtex/2142371c294b5ce2a1750cb3869589496/gdmcbain},
booktitle = {17th AIAA Computational Fluid Dynamics Conference},
doi = {10.2514/6.2005-4622},
eventdate = {6-9 June 2005},
eventtitle = {17th AIAA Computational Fluid Dynamics Conference},
interhash = {d86d7f96562a7b2431c89b57e2b6d871},
intrahash = {142371c294b5ce2a1750cb3869589496},
keywords = {76m28-particle-methods-and-lattice-gas-methods-in-fluid-mechanics smoothed-particle-hydrodynamics},
number = 4622,
publisher = {AIAA},
timestamp = {2023-02-16T04:37:23.000+0100},
title = {An Analysis of Accuracy in One-Dimensional Smoothed Particle Hydrodynamics},
url = {https://arc.aiaa.org/doi/10.2514/6.2005-4622},
venue = {Toronto},
year = 2005
}