Rate of Convegence of a Stochastic Particle Method for the Kolmogorov Equation with Variable Coefficients

Abstract

In a recent paper, E.G. Puckett proposed a stochastic particle method for the non linear diffusion-reaction P.D.E in [0,T] x R (the so-called KPP (Kolmogorov-Petrovskii-Piskunov) equation) :where 1-uo is the cumulative function, supposed to be smooth enough, of a probability distribution, and f 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 L1(= x R) of order 1/NF(1;4), provided h = 0(1/NF(1;4) but conjectured, from numerical experiments, that it should be of order 0(h) + 0(1/A) without any relation between h and N. We prove that conjecture. We also construct a similar stochastic particle method for more general non linear diffusion-reaction-convection P.D.E.'s 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 is in the other case : 0(A) + 0(1/A). The construction of the method and the analysis of the error are based on a stochastic representation formula of the exact solution u.