Nonlocal Stokes equation with relaxation on the divergence free equation

TL;DR

This paper introduces a nonlocal approximation to the Stokes system with a relaxation term, demonstrating well-posedness and convergence to the classical system, thereby enhancing theoretical understanding of related numerical methods.

Contribution

It presents a new nonlocal Stokes model with relaxation, proving its well-posedness and second-order convergence to the local system as the nonlocal interaction horizon shrinks.

Findings

Well-posedness under mild kernel assumptions

Second-order convergence to local Stokes system

Provides theoretical insights for numerical methods like SPH

Abstract

In this paper, we consider a new nonlocal approximation to the linear Stokes system with periodic boundary conditions in two and three dimensional spaces . A relaxation term is added to the equation of nonlocal divergence free equation, which is reminiscent to the relaxation of local Stokes equation with small artificial compressibility. Our analysis shows that the well-posedness of the nonlocal system can be established under some mild assumptions on the kernel of nonlocal interactions. Furthermore, the new nonlocal system converges to the conventional, local Stokes system in second order as the horizon parameter of the nonlocal interaction goes to zero. The study provides more theoretical understanding to some numerical methods, such as smoothed particle hydrodynamics, for simulating incompressible viscous flows.