Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations

Abstract

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 \(\delta \) 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 \(\delta \) approaches zero. For some other commonly used smooth kernels, there are risks in getting ill-posed continuum models that could lead to computational difficulties in practice. This leads us to discuss the implications of our finding on the design of numerical methods.