AUTHOR=Rendón Otto , Avendaño Gilberto D. , Klapp Jaime , Sigalotti Leonardo Di G. , Vargas Carlos A. 

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

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 8 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.907604

DOI=10.3389/fams.2022.907604

ISSN=2297-4687

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 $N\to\infty$ and $h\to 0$, 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 
$\mu _{0}^{N}=\sum_{b=1}^{N}m(x_{b},h)\delta_{0,x_{b}(0)}$, where $m_{b}=m(x_{b},h)$ is
the mass of particle with label $b$ at position $x_{b}(t)$ and $\delta_{0,x_{b}(t)}$ 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 $\mathcal{N}\to\infty$ is
also ensured, where $\mathcal{N}$ is the number of neighbors per particle within the
compact support of the interpolating kernel.