Presentation and analysis of a diffusion-velocity method

¹ Université d'Évry-Val d'Essonne, France
² C.M.A.P., École Polytechnique, France

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) (Chorin A. J.) and the idea was to include most of the effects of the different operators to the dynamics of the particles. Other methods (Rehbach, C.-Fishelov D.) 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 ([Fronteau J.-Huberson S.). This method is based on the idea that the diffusion of a scalar quantity Omega has a prefered direction of transport, namely, that of Nabla Omega. 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.