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 (Formula presented.), where m = m(x, h) is the mass of particle with label b at position x(t) and δ is the x(t)-centered Dirac delta distribution, full consistency of the SPH method is demonstrated in the above joint limit if the additional limit (Formula presented.) → ∞ is also ensured, where (Formula presented.) is the number of neighbors per particle within the compact support of the interpolating kernel.