Convergence of a random walk method for the Burgers equation

Abstract:

We show that the solution of the Burgers equation can be approximated in ${L^1}({\mathbf {R}})$, to within $O({m^{ - 1/4}}{(\ln m)^2})$, by a random walk method generated by $O(m)$ particles. The nonlinear advection term of the equation is approximated by advecting the particles in a velocity field induced by the particles. The diffusive term is approximated by adding an appropriate random perturbation to the particle positions. It is also shown that the corresponding viscous splitting algorithm approximates the solution of the Burgers equation in ${L^1}({\mathbf {R}})$ to within $O(k)$ when k is the size of the time step. This work provides the first proof of convergence in a strong sense, for a random walk method, in which the related advection equation allows for the formation of shocks.