2018/05/21 by Qiang Du, Du, Qiang, Xiaochuan Tian +1
Computer Science · Engineering · Mathematics · #35A23 #45A05 #45P05 #46E35 #FOS: Mathematics #Fluid Dynamics Simulations and Interactions #Lattice Boltzmann Simulation Studies #Numerical Analysis (math.NA) #Numerical methods in engineering #cs.NA #math.NA #msc:35A23 #msc:45A05 #msc:45P05 #msc:46E35
paper · pdf · doi:10.48550/arxiv.1805.08261
modified title, revised form accepted by journal
openalex publication_date 2018/05/21 · arxiv created 2019/07/03 · arxiv updated 2019/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the mathematics of SPH, a nonlocal relaxation to the conventional linear steady state Stokes system for incompressible viscous flows. The nonlocal continuum model is characterized by a smoothing length δ which measures the range of nonlocal interactions. It serves as a bridge between the discrete approximation schemes that involve a nonlocal integral relaxation and the local continuum models. We show that for a class of carefully chosen nonlocal operators, the resulting nonlocal Stokes equation is well-posed and recovers the original Stokes equation in the local limit when δ approaches zero. We also discuss the implications of our finding on the design of numerical methods.