Presentation and analysis of a diffusion-velocity method

Abstract

Lagrangian methods were first introduced to solve purely convective problems approximately. These problems include the compressible Euler equations in fluid mechanics, the Vlasov equation in plasma physics. These methods have been extended to non-purely convective problems, to take into account diffusion effects. This has been first done by addition of a random walk to the movement of particles (inclusion of a brownian effect) [1] and the idea was to include most of the effects of the different operators to the dynamics of the particles. Other methods ([2]-[7]) appeared with a different way ot thinking of particles as carriers of information, not only through their position as in the case of random walk, but also through their coefficient. Allowing not only the position but also the weight of the particles to evolve in time, has permitted particle methods to tackle more general operators. Indeed, particles are then no longer considered as clouds of independent points but rather as a set of interacting points.More recently a new method has emerged: the diffusion-velocity method ([8]-[14]). This method is based on the idea that the diffusion of a scalar quantity ω has a prefered direction of transport, namely, that of ∇ω. This remark results in the transformation of a diffusion equation into an advection equation.In the first section, we present the diffusion-velocity method in the linear case and prove an existence and uniqueness result for a finite time. Additional results are then given for the 1-d case in the second section. In the third section, we consider the case of the 2-d, then 3-d, Navier-Stokes equations. The linear caseConsider the linear advection-diffusion equation on the whole n-dimensional space with initial condition:We assume from now on that the initial condition ω 0 is positive. This implies of course that the solution ω is also positive. In this case, by settingequation (1) can be rewritten as a purely transport equation:Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx

Particle Methods for Dispersive Equations

“…The proof follows the arguments of Lacombe and MasGallic [25] for the diffusion-velocity transport equation (see also [24]). Here, however, we improve the result of [25] by observing that the resulting solution has the same regularity as the initial data.…”

“…The proof follows the arguments of [25] with the required adaptations to the dispersive framework and additional bootstrapping arguments regarding the regularity of the solution. It is based on a fixed point argument on the functional …”

“…The main analytical result in this section is Theorem 2.1, where we prove (in the spirit of [25]) a short time existence and uniqueness for solutions of the dispersion-velocity transport equation. This theorem requires the initial data to have only one bounded derivative and provides the same regularity for the resulting solution.…”

“…Particles carrying fixed masses will be then convected with speed a(u). The convergence properties of the diffusion-velocity method were investigated, e.g., in [24,25], where short time existence and uniqueness of solutions to the resulting diffusion-velocity transport equation were proved. The diffusion-velocity method serves as the basic tool for the derivation of our particle methods in the dispersive world.…”

Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations

“…It was first introduced by Fronteau and Combis (1984) and popularized by ; Degond and Mustieles (1990), Ogami and Akamatsu (1991) and Kempka and Strickland (1993) in the early 1990s. This method was then largely analysed (Strickland et al 1996;Mas-Gallic 1999;Lacombe and Mas-Gallic 1999;Lacombe 1999;Lions and Mas-Gallic 2001), adapted to dispersion equations (Lions and Mas-Gallic 2001;Levy 2001, 2002), coupled with turbulence models (Milane 2004) and extended to the diffusion of a vector field and to axisymmetric flows (Rivoalen et al 1997;Grant and Marshall 2005;Rivoalen and Huberson 1999).…”