[Paper Review] A Study of the Stability Properties of SPH

Joseph P. Morris|arXiv (Cornell University)|Apr 3, 1995

Fluid Dynamics Simulations and InteractionsEngineering22 citations

TL;DR

This paper investigates instabilities in Smoothed Particle Hydrodynamics (SPH) arising from momentum-conserving formulations when stress becomes negative, such as in MHD or incompressible flows. It demonstrates that higher-order spline kernels—particularly those approximating a Gaussian—significantly improve stability by suppressing spurious transverse modes and reducing growth rates of instabilities, while alternative formulations using pressure differences avoid these issues altogether.

ABSTRACT

When using a formulation of Smooth Particle Hydrodynamics (SPH) which conserves momentum exactly the motion of the particles is observed to be unstable to negative stress. It is also found that under normal circumstances a lattice of SPH particles is potentially unstable to transverse waves. This document is a summary of a detailed report (Morris 1994) investigating the nature of these and other instabilities in depth. Approaches which may be used to eliminate these instabilities are suggested. It is found that the stability properties of SPH in general improve as higher order spline interpolants, approximating a Gaussian, are used as kernels.

Motivation & Objective

• To identify the root cause of numerical instabilities in momentum-conserving SPH formulations when stress becomes negative.
• To analyze the stability of SPH under various kernel functions, especially in one-dimensional and lattice-based configurations.
• To evaluate how kernel choice—particularly higher-order splines approximating a Gaussian—affects the growth rates of unstable modes.
• To propose alternative formulations that maintain momentum conservation while avoiding instability, especially in problems with negative or adjustable background pressure.
• To provide guidelines for selecting stable, accurate SPH kernels and formulations for complex fluid dynamics problems, including MHD and incompressible flows.

Proposed method

• Derives the linearized dispersion relation for one-dimensional SPH using symmetric kernels, including Gaussian and cubic spline, to analyze wave propagation and stability.
• Applies Poisson summation formula to transform the discrete particle sum into a Fourier series for stability analysis across periodic lattices.
• Compares two formulations: the standard momentum-conserving form (sensitive to background pressure) and an alternative using pressure differences (independent of background pressure).
• Uses numerical analysis to compute growth rates of unstable transverse modes in rectangular and hexagonal particle lattices.
• Evaluates the impact of kernel compact support and Fourier transform decay on instability growth, showing faster decay (e.g., Gaussian-like kernels) suppresses instabilities.
• Proposes splitting the stress tensor into a positive component (handled via momentum-conserving SPH) and a remainder via differencing to maintain stability.

Experimental results

Research questions

• RQ1 Why does momentum-conserving SPH become unstable under negative stress, and what is the fundamental mechanism behind this instability?
• RQ2 How do different kernel functions—especially higher-order splines approximating a Gaussian—affect the stability of SPH in one-dimensional and lattice-based flows?
• RQ3 Can alternative formulations that avoid action-reaction pair forces in momentum equations eliminate instabilities caused by negative stress?
• RQ4 What role does the Fourier transform decay rate of the kernel play in suppressing unstable transverse modes in SPH?
• RQ5 How do particle lattice geometry (rectangular vs. hexagonal) and dimensionality (1D, 2D, 3D) influence the onset and growth of instabilities?

Key findings

• The standard momentum-conserving SPH formulation becomes unstable under negative stress due to particle pairing and clumping, driven by particles falling into mutual potential wells.
• Instabilities in transverse modes are strongly suppressed when higher-order spline kernels—especially those approximating a Gaussian—are used, with negligible growth rates for practical smoothing lengths.
• The pressure-difference formulation (Eq. 9) eliminates dependence on background pressure and avoids the instability entirely, though it no longer conserves momentum exactly.
• Stability improves significantly with kernels whose Fourier transforms decay rapidly, as seen in Gaussian and high-order spline kernels, reducing spurious mode growth.
• Rectangular and hexagonal lattices both exhibit unstable transverse modes with compact-support kernels, but growth rates drop dramatically with higher-order kernels.
• For complex equations of state or strong shocks, splitting the stress tensor into a positive component and a remainder via differencing is a viable strategy to maintain stability while preserving accuracy.