A research on the tensile instability of SPH in fluid dynamics

A small perturbation and linearization method have been employed to analyze the tensile instability of Smoothed Particle Hydrodynamics (SPH), after the matrix form of SPH is introduced. With the former method, the propagation of numerical errors in SPH are found, which consists of the routine error propagation and the distortion of errors' phases. With the latter method, the errors' linear form and its systematical matrix are obtained. The sufficient condition for tensile instability from Swegle could be obtained through both methods above. Omitting the effects from continuity and constitutive equations, the characteristic equation of the systematical matrix is derived. Thus a representative matrix which could approximately indicate the existence of tensile instability is derived. The eigenvalues of this representative matrix are calculated for a system with 3-nearby particles. These eigenvalues transit from saddle points, central points to focus points if differences exist among the errors' phases. Under the request for eigenvalues to stabilize the error system of SPH at the wave number K=φ/ΔX, appropriate initial smoothing lengths for different smoothing functions are derived. And a relation among the numerical sound speed in weakly-incompressible method, smoothing length refreshing and the stability of SPH is found, and which could theoretically give the values of the density variation rate from Monaghan and Morris' numerical researches.