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Rate of convergence of a stochastic particle method for the Kolmogorov equation with variable coefficients

Authors: Pierre Bernard, Denis Talay and Luciano Tubaro
Journal: Math. Comp. 63 (1994), 555-587, S11
MSC: Primary 65M12; Secondary 35K57, 60J15, 60J60
MathSciNet review: 1250770
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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):

$\displaystyle \left\{ \begin{array}{*{20}{c}} \frac{{\partial u}}{{\partial t}}... ... + f(u), \hfill \\ u(0, \cdot ) = {u_0}( \cdot ), \hfill \\ \end{array} \right.$

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

$\displaystyle \left\{ \begin{array}{*{20}{c}} \frac{{\partial u}}{{\partial t}} = Lu + f(u), \hfill \\ u(0,\cdot) = {u_0}(\cdot), \hfill \\ \end{array} \right.$

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.

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Article copyright: © Copyright 1994 American Mathematical Society