On the use of circulant matrices for the stability analysis of recent weakly compressible SPH methods

TL;DR

This paper uses circulant matrices and Fourier analysis to derive stability conditions for weakly compressible SPH methods, revealing how background pressure influences stability and wave propagation accuracy.

Contribution

It introduces a novel stability analysis framework using circulant matrices and Fourier transforms for WCSPH, providing analytical insights into stability and wave behavior.

Findings

No tensile or pairing instabilities found.

Optimal background pressure for wave propagation is around ρc^2.

Viscous damping matches continuum predictions at p_back ≈ ρc^2/2.

Abstract

In this study, a linear stability analysis is performed for different Weakly Compressible Smooth Particle Hydrodynamics (WCSPH) methods on a 1D periodic domain describing an incompressible base flow. The perturbation equation can be vectorized and written as an ordinary differential equation where the coefficients are circulant matrices. The diagonalization of the system is equivalent to apply a spatial discrete Fourier transform. This leads to stability conditions expressed by the discrete Fourier transform of the first and second derivatives of the kernel. Although spurious modes are highlighted, no tensile nor pairing instabilities are found in the present study, suggesting that the perturbations of the stresses are always damped if the base flow is incompressible. The perturbations equation is solved in the Laplace domain, allowing to derive an analytical solution of the transient state. Also, it is demonstrated analytically that a positive background pressure combined with the uncorrected gradient operator leads to a reordering of the particle lattice. It is also shown that above a critical value, the background pressure leads to instabilities. Finally, the dispersion curves for inviscid and viscous flows are plotted for different WCSPH methods and compared to the continuum solution. It is observed that a background pressure equal to $\rho c^2$ gives the best fidelity to predict the propagation of a sound wave. When viscosity effects are taken into account, the damping of pressure fluctuations show the best agreement with the continuum for $p_{back} \sim \rho c^2/2$.