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

DOI:
https://doi.org/10.1090/S0025-5718-1994-1250770-3

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 (the so-called "KPP" (Kolmogorov-Petrovskii-Piskunov) equation):

*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 in of order , provided , but conjectured, from numerical experiments, that it should be of order , 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

*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 .

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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1994-1250770-3

Article copyright:
© Copyright 1994
American Mathematical Society