Large-Time Behavior of Deterministic Particle Approximations to the Navier-Stokes Equations

Abstract

Abstract.We prove that for a class of deterministic vortex methods for the Navier-Stokes equations in two dimensions, the numerical solution decays for large time with the same rate as the exact solution. We substantiate our result with numerical experiments and with a remark concerning the problem of reinitialization of a distribution of particles.

Vortex particle-in-cell method for three-dimensional viscous unbounded flow computations

“…Several procedures have been derived to model viscous diffusion, such as the random walk of Chorin [2] and the deterministic method of Cottet [26]. The consistency and convergence of these schemes is addressed in papers by, among others, Hald [25] and Hou [27].…”

“…For a Gaussian vorticity distribution in the core of the ring, the constant C is equal to 0.558. From Equation (26), the effect of viscosity is to slow down the motion of the ring. As we already noted, the propagation velocity depends on the core radius, and Equation (26) accounts for the classical wt viscous spreading.…”

“…However, if we consider the effect of viscosity, i.e. use Equation (26), then the 'effective core radius' | is equal to 0.0097. The mesh size used in the simulation was 0.0156, therefore, the mesh radius is 0.0078.…”

Convergence of high-order deterministic particle methods for the convection-diffusion equation

“…The size of the domain where the initial particles are placed is often estimated from the final time of the specific simulation. Cottet [5] has made a study of the large-time behavior of the PSE method (with high-order kernel) for the vorticity formulation of Navier-Stokes in two dimensions. He proved that the numerical solution exhibits the same decay rate (of enstrophy) as the solution of the continuous problem for large times.…”