TY - JOUR

T1 - Dispersion analysis of SPH for parabolic equations: High-order kernels against tensile instability

AU - Stoyanovskaya, O. P.

AU - Burmistrova, O. A.

AU - Arendarenko, M. S.

AU - Markelova, T. V.

N1 - Сведения о финансировании Финансирующий спонсор Номер финансирования Российский научный фонд 23-11-00142

PY - 2025/3/15

Y1 - 2025/3/15

N2 - The Smoothed Particle Hydrodynamics (SPH) is a meshless particle-based method mainly used to solve dynamical problems for partial differential equations (PDE). By means of dispersion analysis we investigated four classical SPH-discretizations of parabolic PDE differing by the approximation of Laplacian. We derived approximate dispersion relations (ADR) for considered SPH-approximations of the Burgers equation. We demonstrated how the analysis of the ADR allows both studying the approximation and stability of numerical scheme and explaining the features of the method that are known from practice, but are counter-intuitive from the theoretical point of view. By means of the mathematical analysis of ADR, the phenomenon of conditional approximation of some schemes under consideration is shown. Moreover, we pioneered in obtaining the necessary condition for the stability of the SPH-approximation of parabolic equations in terms of the Fredholm integral operator applied to the function defined by the kernel of the SPH method. Using this condition, we revealed that passing from the classical second-order kernels to high-order kernels for some schemes leads to the appearance of tensile (short-wave) instability. Among the schemes under consideration, we found the one, for which the necessary condition for the stability of short waves is satisfied both for classical and high-order kernels. The fourth order of approximation in space of this scheme is shown theoretically and confirmed in practice.

AB - The Smoothed Particle Hydrodynamics (SPH) is a meshless particle-based method mainly used to solve dynamical problems for partial differential equations (PDE). By means of dispersion analysis we investigated four classical SPH-discretizations of parabolic PDE differing by the approximation of Laplacian. We derived approximate dispersion relations (ADR) for considered SPH-approximations of the Burgers equation. We demonstrated how the analysis of the ADR allows both studying the approximation and stability of numerical scheme and explaining the features of the method that are known from practice, but are counter-intuitive from the theoretical point of view. By means of the mathematical analysis of ADR, the phenomenon of conditional approximation of some schemes under consideration is shown. Moreover, we pioneered in obtaining the necessary condition for the stability of the SPH-approximation of parabolic equations in terms of the Fredholm integral operator applied to the function defined by the kernel of the SPH method. Using this condition, we revealed that passing from the classical second-order kernels to high-order kernels for some schemes leads to the appearance of tensile (short-wave) instability. Among the schemes under consideration, we found the one, for which the necessary condition for the stability of short waves is satisfied both for classical and high-order kernels. The fourth order of approximation in space of this scheme is shown theoretically and confirmed in practice.

KW - Approximate dispersion relation

KW - Burgers equation

KW - Dispersion analysis

KW - Fourier analysis

KW - High-order SPH method

KW - Spectral analysis

KW - Tensile instability

KW - Von Neumann analysis

KW - Wendland kernel

UR - https://www.mendeley.com/catalogue/a48ccdd2-e795-35be-ac90-7dd23c258abb/

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85206889393&origin=inward&txGid=ffedc804b74dbade131283313b602545

U2 - 10.1016/j.cam.2024.116316

DO - 10.1016/j.cam.2024.116316

M3 - статья

VL - 457

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 116316

ER -