In this paper we construct a stochastic particle method for the
Burgers equation with a monotone initial condition; we prove that the
convergence rate is $O(1/ \sqrt{N} + \sqrt{\Delta t})$ for the $L^1 (\mathbb{R}
\times \Omega)$ norm of the error. To obtain that result, we link the PDE and
the algorithm to a system of weakly interacting stochastic particles; the
difficulty of the analysis comes from the discontinuity of the interaction
kernel, which is equal to the Heaviside function.
¶ In a previous paper we showed how the algorithm and the result
extend to the case of nonmonotone initial conditions for the Burgers equation.
We also treated the case of nonlinear PDE's related to particle systems with
Lipschitz interaction kernels. Our next objective is to adapt our methodology
to the (more difficult) case of the two-dimensional inviscid Navier-Stokes
equation.
@article{1034968229,
author = {Bossy, Mireille and Talay, Denis},
title = {Convergence rate for the approximation of the limit law of weakly
interacting particles: application to the Burgers equation},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 818-861},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968229}
}
Bossy, Mireille; Talay, Denis. Convergence rate for the approximation of the limit law of weakly
interacting particles: application to the Burgers equation. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 818-861. http://gdmltest.u-ga.fr/item/1034968229/