TY - JOUR

T1 - Dispersion analysis of SPH as a way to understand its order of approximation

AU - Stoyanovskaya, O. P.

AU - Lisitsa, V. V.

AU - Anoshin, S. A.

AU - Savvateeva, T. A.

AU - Markelova, T. V.

N1 - The study was founded by RSF grant 21-19-00429 (asymptotic analysis done by T.M., computational experiments done by O.S.), NSU (visualization of results done by S.A. and T.S.) and IPGG state funding of the research project no. FWZZ-2022-0022 (dispersion analysis for fully discrete form done by V.L.), LIH state funding of the research project no. FWZZ-2021-0001 (dispersion analysis for semi-discrete form done by T.S.).

PY - 2024/3/1

Y1 - 2024/3/1

N2 - 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 h, the distance between the particles Δx. 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 Δx/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 [Formula presented], where ξ is a parameter determined by the shape of kernel (its smoothness, i.e. the number of continuum derivatives), η 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 Δx/h. To this end kernels of which smoothness is at least 4 are necessary.

AB - 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 h, the distance between the particles Δx. 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 Δx/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 [Formula presented], where ξ is a parameter determined by the shape of kernel (its smoothness, i.e. the number of continuum derivatives), η 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 Δx/h. To this end kernels of which smoothness is at least 4 are necessary.

KW - Dispersion analysis

KW - Dispersion relation

KW - Fourier analysis

KW - High-order SPH

KW - Order of approximation

KW - Smoothed particle hydrodynamics (SPH)

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85170096046&origin=inward&txGid=8f98d8f829fb6fa7e591c48af50bc18f

UR - https://www.mendeley.com/catalogue/290722f9-7700-3c38-91dd-469f12d746bb/

U2 - 10.1016/j.cam.2023.115495

DO - 10.1016/j.cam.2023.115495

M3 - Article

VL - 438

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 115495

ER -