Local Consistency of Smoothed Particle Hydrodynamics (SPH) in the Context of Measure Theory - Rendón, Otto; Avendaño, Gilberto D.; Klapp, Jaime; Sigalotti, Leonardo Di G.; Vargas, Carlos A. (Frontiers in Applied Mathematics and Statistics, 2022)
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 filling the computational domain and $h$ is the smoothing length. Using an initial local discrete measure given by $μ _{0}^{N}=\sum_{b=1}^{N}m(x_{b},h)δ_{0,x_{b}(0)}$, where $m_{b}=m(x_{b},h)$ is the mass of particle with label $b$ at position $x_{b}(t)$ and $δ_{0,x_{b}(t)}$ is the $x_{b}(t)$-centered Dirac delta distribution, full consistency of the SPH method is demonstrated in the above joint limit if the additional limit $\mathcal{N}→∞$ is also ensured, where $\mathcal{N}$ is the number of neighbors per particle within the compact support of the interpolating kernel.