Convergence speed of stochastic particle algorithms and application to the Burgers equation.

Summary The convergence of the random vortex method for the navier-stokes equation has not yet been established in a fully satisfactory sense. This problem has strongly motivated the study of particle algorithms for some nonlinear P.D.E.'s, in particular, the burger equation which we present in this paper. The objective of this work is to give new results of convergence speed of stochastic particle methods, using the probabilistic interpretation of nonlinear e. D. P in terms of a system of interacting particles. The theory of stochastic processes allows us to interpret nonlinear e. D. P of the mckean-vlasov type as limit equations for systems of interacting particles. We derive a simple and natural algorithm based on the simulation of the underlying particle system. We obtain the speed of convergence of the algorithm, when the interaction kernels are lipschitzian and bounded. We then give a new probabilistic interpretation of the Burgers equation in terms of a system of interacting particles (the corresponding interaction kernel is discontinuous) and show that the system of particles possesses the propagation property of chaos. We study the convergence (theoretical and numerical) of the algorithm. The convergence speed we obtain seems to be what can be expected for this family of algorithms and gives a new theoretical insight to the random vortex method.