Local Consistency of Smoothed Particle Hydrodynamics (SPH) in the Context of Measure Theory

The local consistency of the method of Smoothed Particle Hydrodynamics (SPH) is proved for a multidimensional continuous mechanical system in the context of measure theory. The Wasserstein distance of the corresponding measure-valued evolutions is used to show that full convergence is achieved in the joint limit N → ∞ and h → 0, where N is the total number of particles that discretize the computational domain and h is the smoothing length. Using an initial local discrete measure given by μN0=∑Nb=1m(xb,h)δ0,xb(0)μ0N=∑b=1Nm(xb,h)δ0,xb(0), where mb = m(xb, h) is the mass of particle with label b at position xb(t) and δ0,xb(t) is the xb(t)-centered Dirac delta distribution, full consistency of the SPH method is demonstrated in the above joint limit if the additional limit NN → ∞ is also ensured, where NN is the number of neighbors per particle within the compact support of the interpolating kernel.