Frontiers in Applied Mathematics and Statistics (Jun 2022)
Local Consistency of Smoothed Particle Hydrodynamics (SPH) in the Context of Measure Theory
Abstract
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 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 N → ∞ is also ensured, where N is the number of neighbors per particle within the compact support of the interpolating kernel.
Keywords
