A Cellular Automaton for Burgers' Equation

A bstract. We study the app roximation of solut ions to the Burgers' equation , an + c!.-(n _n 2) = v a 2 n ilt ilx 2 ilx' (1) by spatially avera ging a prob abilistic cellular automaton motivated by random walks on a line. The auto maton consists of moving "particles" on Q. one-dimensional periodic lattice with spe ed one and in a random direction subject to the exclusion principle that at most one particle may move in a given direction from a given lattice site, at a given t ime. The exclusion principle gives rise to th e nonlinearity in Eq. (1) and int roduces corr elations between the particles which must be est imated t o obtain statistical bounds on th e error. These bounds are obtained in two steps. The first is showing th at th e ensemble average of the automaton is a stable exp licit finite differencing scheme of Eq. (1) over th e lattice with a second order convergence in the lat tice spaci ng. The numerical diffusion of this scheme play s an important role in relating the automaton rules to Eq. (1). The next st ep is showing t hat the spatial averaging of a single evolution of the automaton conver ges to the spatial averag ing of the ensemble as I/ VMwhere M is the number of lat t ice sit es averaged. Simulations are presented and discussed. 1. Int r oduction Recently it has b een p roposed to use ce llu la r au tomat a on large lat ti ces for obtaining sol ut ions t o parti al differ ential equations, in particula r the incompress ib le Navier-S tokes equat ions [11. Suc h a ut om at a h ave rules with locally conserved (or nearly conserve d) qu anti ti es whi ch, when averaged