Rate of convergence 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 nonlinear diffusion-reaction PDE in $[0,T] \times \mathbb {R}$ (the so-called "KPP" (Kolmogorov-Petrovskii-Piskunov) equation): \[ \left \{ \begin {array}{*{20}{c}} \frac {{\partial u}}{{\partial t}} = Au = \Delta u + f(u), \hfill \\ u(0, \cdot ) = {u_0}( \cdot ), \hfill \\ \end {array} \right .\] where $1 - {u_0}$ 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 ${L^1}(\Omega \times \mathbb {R})$ of order $1/{N^{1/4}}$, provided $h = \mathcal {O}(1/{N^{1/4}})$, but conjectured, from numerical experiments, that it should be of order $\mathcal {O}h + \mathcal {O}(1/\sqrt N )$, 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 \[ \left \{ \begin {array}{*{20}{c}} \frac {{\partial u}}{{\partial t}} = Lu + f(u), \hfill \\ u(0,\cdot ) = {u_0}(\cdot ), \hfill \\ \end {array} \right .\] 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 $\mathcal {O}(\sqrt h ) + \mathcal {O}(1/\sqrt N )$. The construction of the method and the analysis of the error are based on a stochastic representation formula of the exact solution u.