# American Institute of Mathematical Sciences

September  2017, 10(3): 613-641. doi: 10.3934/krm.2017025

## Numerical study of a particle method for gradient flows

 1 Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK 2 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK 3 Mathematics Department, Technion--Israel Institute of Technology, Haifa 32000, Israel

* Corresponding author: J. A. Carrillo

Received  June 2016 Revised  October 2016 Published  December 2016

Fund Project: The first, second and third authors are supported by Engineering and Physical Sciences Research Council grant EP/K008404/1. The first author is also supported by the Royal Society through a Wolfson Research Merit Award. The last author is supported by ISF grant 998/5.

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

Citation: José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic and Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025
##### References:

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##### References:
The heat equation
The porous medium equation with $m=2$
The linear Fokker-Planck equation with $\Delta t = 10^{-5}$ --Stabilisation in time of the scheme (rate of convergence to the discrete steady state)
The nonlinear Fokker-Planck equation with $m=2$ and $\Delta t = \frac{0.1}{N^2}$ -Stabilisation in time of the scheme (rate of convergence to the discrete steady state)
The particles' positions for the two-dimensional heat equation for $N=100$ with $\Delta t = 10^{-4}$
Accuracy for the two-dimensional heat equation with $\Delta t = 10^{-4}$
The modified Keller-Segel equation with $\chi = 1.5$ for $N=50$
The blow-up of the modified Keller-Segel equation with $\chi = 1.5$
Blow-up formation for the modified Keller-Segel equation with two initial Gaussian bumps
The modified nonlinear Keller-Segel equation with $\chi = 1.4$ for different choices of $m$, for $N=50$ at $T=4$ with $\Delta t = 10^{-5}$
Compactly supported potential $W(x) = -c\max(1-|x|,0) + c$ with nonlinear diffusion with $c = 8$ and $m = 3$, for $N=80$ with $\Delta t = 10^{-5}$

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