Truncation error in mesh‐free particle methods

By Nathan J. Quinlan, Mihai Basa, Martin Lastiwka. 2006

302 citations recorded by the source corpus.

A truncation error analysis has been developed for the approximation of spatial derivatives in smoothed particle hydrodynamics (SPH) and related first‐order consistent methods such as the first‐order form of the reproducing kernel particle method. Error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length, Δx/h. For uniformly spaced particles in one dimension, analysis shows that as h is reduced while maintaining constant Δx/h, error decays as h2 until a limiting discretization error is reached, which is independent of h. If Δx/h is reduced while maintaining constant h (i.e. if the number of neighbours per particle is increased), error decreases at a rate which depends on the kernel function's smoothness. When particles are distributed non‐uniformly, error can grow as h is reduced with constant Δx/h. First‐order consistent methods are shown to remove this divergent behaviour. Numerical experiments confirm the theoretical analysis for one dimension, and indicate that the main results are also true in three dimensions. This investigation highlights the complexity of error behaviour in SPH, and shows that the roles of both h and Δx/h must be considered when choosing particle distributions and smoothing lengths. Copyright © 2005 John Wiley & Sons, Ltd.

Subfields: Civil and Structural Engineering, Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/50a0956cfd1b6cac717f6a7cd4cd8f4c6401264a

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Citations in the corpus

Permeable and non‐reflecting boundary conditions in SPH (2009, methodology): “the number of neighbour particles is low) [7].”
A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves (2016, background): “Quinlan et al. (2006) discussed the theoretical convergence of approximating the gradient of a function used in SPH.”
Shocking results without shocks: Subsonic turbulence in smoothed particle hydrodynamics and moving-mesh simulations (2011, methodology): “However, SPH’s equation of motion has a ‘zeroth-order’ error in the pressure gradient which is proportional to the pressure itself (e.g. Quinlan et al. 2006; Read et al. 2010; Gaburov & Nitadori 2011; Amicarelli et al. 2011).”
Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future (2016, background): “Details about the effect of particle disorder can be found in Quinlan, Basa, and Lastiwka (2006) and Amicarelli, Marongiu, Leboeuf, Leduc, and Caro (2011).”
A velocity--diffusion method for a Lotka--Volterra system with nonlinear cross and self-diffusion (2009, unknown): “[11] and [21].”
Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method (2008, background): “Further effects that reduce the order are for instance the irregular distribution of the particles (as opposed to the initial square lattice distribution [1]) and the ratio of smoothing length to particle spacing, see [18] for instance.”
Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves (2012, background): “[23] and Schwaiger [26] that the disordered particle distribution will also complicate the convergence behaviour of operators.”