Rate of Convergence of a Stochastic Particle Method

In a recent paper, E. G. Puckett proposed a stochastic particle method for the nonlinear diffusion-reaction PDE in [0, T]xR (the so-called "KPP" (Kolmogorov-Petrovskii-Piskunov) equation): %=Au = Au + f(u), u(0, •) = «(,(•), where 1 Mr, is the cumulative function, supposed to be smooth enough, of a probability distribution, and / is a function describing the reaction. His justification of the method and his analysis of the error were based on a splitting of the operator A . He proved that, if h is the time discretization step and N the number of particles used in the algorithm, one can obtain an upper bound of the norm of the random error on u(T, x) in L'(^XK) of order l/TV1/4, provided h = ¿^(l/JV1/4), but conjectured, from numerical experiments, that it should be of order tf (h) + cf (l/y/Ñ), without any relation between h and N. We prove that conjecture. We also construct a similar stochastic particle method for more general nonlinear diffusion-reaction-convection PDEs ft=Lu + f(u), u(0, •) = "<)(•), where L is a strongly elliptic second-order operator with smooth coefficients, and prove that the preceding rate of convergence still holds when the coefficients of L are constant, and in the other case is cf(yfh) + cf (l/y/Ñ). The construction of the method and the analysis of the error are based on a stochastic representation formula of the exact solution u .