von Neumann stability analysis of smoothed particle hydrodynamics—suggestions for optimal algorithms

TLDR

SPH is a unique particle‑based method, and we compare it with well‑used high‑resolution finite‑difference schemes. We conduct a von Neumann stability analysis of SPH equations to identify optimal parameter ranges for the algorithm. The analysis models the full one‑dimensional Euler equations without major restrictions, enabling derivation of optimal parameter ranges. The study finds that for the M5 spline the smoothing‑length to interparticle‑distance ratio should be 1.0–1.4, that linear artificial viscosity and spatial‑filtering coefficients must be bounded, and offers suggestions for future algorithm improvements.

Abstract

We present a von Neumann stability analysis of the equations of smoothed particle hydrodynamics (SPH) along with a critical discussion of various parts of the algorithm. The stability analysis is done without any major restrictions and, hence, models the full Euler equations in one dimension. This then allows us to deduce optimal ranges for parameters that need to be used in SPH. Thus we show that for the commonly used M5 spline the ratio of smoothing length to interparticle distance should range between 1.0 to 1.4. We also show that the linear artificial viscosity coefficient and the coefficient of spatial filtering have to be bounded. The results of this von Neumann stability analysis provide us with several suggestions for future algorithm improvement. Because the SPH method is so unique we provide, wherever possible, comparisons with more familiar and well-used high resolution finite difference methods.