TY - JOUR
T1 - From continuum mechanics to SPH particle systems and back
T2 - systematic derivation and convergence
AU - Evers, Joep H.M.
AU - Zisis, Iason A.
AU - van der Linden, Bas J.
AU - Duong, Manh Hong
PY - 2018/1/1
Y1 - 2018/1/1
N2 - In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method [27], and with the equation treated by Di Lisio et al. in [9], respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.
AB - In this paper, we derive from the principle of least action the equation of motion for a continuous medium with regularized density field in the context of measures. The eventual equation of motion depends on the order in which regularization and the principle of least action are applied. We obtain two different equations, whose discrete counterparts coincide with the scheme used traditionally in the Smoothed Particle Hydrodynamics (SPH) numerical method [27], and with the equation treated by Di Lisio et al. in [9], respectively. Additionally, we prove the convergence in the Wasserstein distance of the corresponding measure-valued evolutions, moreover providing the order of convergence of the SPH method. The convergence holds for a general class of force fields, including external and internal conservative forces, friction and non-local interactions. The proof of convergence is illustrated numerically by means of one and two-dimensional examples.
KW - convergence rate
KW - measure-valued equations
KW - principle of least action
KW - Smoothed Particle Hydrodynamics
KW - Wasserstein distance
UR - http://www.scopus.com/inward/record.url?scp=85040785957&partnerID=8YFLogxK
U2 - 10.1002/zamm.201600077
DO - 10.1002/zamm.201600077
M3 - Article
AN - SCOPUS:85040785957
VL - 98
SP - 106
EP - 133
JO - Zeitschrift für Angewandte Mathematik und Mechanik
JF - Zeitschrift für Angewandte Mathematik und Mechanik
SN - 0044-2267
IS - 1
ER -