An analysis of accuracy in one-dimensional Smoothed Particle Hydrodynamics

A truncation error analysis has been developed for the approximation of spatial derivatives in one-dimensional Smoothed Particle Hydrodynamics (SPH). For this purpose, the SPH interpolation is understood as a two-step process of smoothing and discretisation. As smoothing length is reduced while maintaining a constant ratio of particle spacing Δx to smoothing length h, error decays as h²; however, there is a finite limiting discretisation error. If particle spacing Δx is reduced while holding a constant h, error decreases at a rate which depends on the kernel function's smoothness at its boundaries. When particles are distributed non-uniformly, error behaviour is complex, and discretisation error can diverge as h is reduced. A first-order consistent SPH method is shown to remove this behaviour. The findings of the theoretical analysis are confirmed by numerical experiments.