Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Discrete-in-time random particle blob method for the Keller–Segel equation and convergence analysis

Pages: 1821 – 1842

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n7.a2

Authors

Hui Huang (Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada)

Jian-Guo Liu (Departments of Physics and Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Abstract

We establish an error estimate of a discrete-in-time random particle blob method for the Keller–Segel (KS) equation in $\mathbb{R}^d (d \geq 2)$. With a blob size $\epsilon=N^{- \frac{1}{d(d+1)}} \log(N)$, we prove the convergence rate between the solution to the KS equation and the empirical measure of the random particle method under $L^2$ norm in probability, where $N$ is the number of the particles.

Keywords

coupling method, concentration inequality, splitting scheme, kernel density estimation, Newtonian aggregation, chemotaxis, Brownian motion, interacting particle system

2010 Mathematics Subject Classification

35K55, 35Q92, 60H35, 65M12, 65M15, 65M75

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Hui Huang is partially supported by the Alan Mekler Postdoctoral Fellowship in the Department of Mathematics at Simon Fraser University.

Jian-Guo Liu is partially supported by KI-Net NSF RNMS (Grant No. 1107444) and NSF DMS (Grant No. 1514826).

Received 3 April 2016

Accepted 15 April 2017

Published 16 October 2017