Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Rate of convergence of a stochastic particle method for the Kolmogorov equation with variable coefficients
HTML articles powered by AMS MathViewer

by Pierre Bernard, Denis Talay and Luciano Tubaro PDF
Math. Comp. 63 (1994), 555-587 Request permission

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.
References
Similar Articles
Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Math. Comp. 63 (1994), 555-587
  • MSC: Primary 65M12; Secondary 35K57, 60J15, 60J60
  • DOI: https://doi.org/10.1090/S0025-5718-1994-1250770-3
  • MathSciNet review: 1250770