Abstract: Smoothed Particle Hydrodynamics (SPH) is a numerical method to solve dynamical partial differential equations (PDE). The basis of the method is a << kernel-based>> way to compute the spatial derivatives of a function whose values are given in moving irregularly located nodes (Lagrangian particles). Accuracy of the SPH is determined by independent parameters — the shape of the kernel, the kernel size, the distance between the particles ∆x. Constructing high-order SPH-schemes for different types of PDE is a state-of-the-art problem of computational mathematics. For the classical SPH-approximation of one-dimensional hyperbolic equations (isothermal gas dynamics) we found that the order of approximation of smooth solution correlates to the dispersion properties of the method. To this end we analyzed the dispersion relation for the approximation and found analytical representation of the numerical wave phase velocity. Moreover, for the first time, the order of approximation with respect to ∆x/h was confirmed in computational experiments on a dynamic problem of sound wave propagation. For two kernels with 2 and 4 continuum derivatives, the second and the fourth order of approximation, respectively, was found. This finding may be generalized as follows. The solution error in the one-dimensional case for a quasi-uniformly located particles has the form O((h λ)η+(∆x h )ξ), where ξ is a parameter determined by the shape of kernel (its smoothness, i.e. the number of continuum derivatives), is a parameter that does not depend on the shape of kernel (for classical non-negative kernels η=2), λ is the wavelength. Our results indicates that to develop high-order SPH-schemes for hyperbolic equations besides improving the order of approximation with respect to h/λ one need to ensure the order of approximation with respect to ∆x/h. To this end kernels of which smoothness is at least 4 are necessary.

Cite: Stoyanovskaya O.P. , Lisitsa V.V. , Anoshin S.A. , Savvateeva T.A. , Markelova T.V.
Dispersion Analysis of SPH as a Way to Understand Its Order of Approximation
Journal of Computational and Applied Mathematics. 2024. V.438. 115495 :1-15. DOI: 10.1016/j.cam.2023.115495 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Mar 9, 2023
Accepted:	Jul 23, 2023
Published online:	Aug 12, 2023
Published print:	Mar 1, 2024

Identifiers:

Web of science:	WOS:001080114900001
Scopus:	2-s2.0-85170096046
Elibrary:	64862089
OpenAlex:	W4385776256

Citing:

DB	Citing
Scopus	5
OpenAlex	3
Web of science	1
Elibrary	4

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