Numerical Study of a Particle Method for Gradient Flows

TL;DR

This paper investigates a particle-based numerical method for gradient flows with diffusion, demonstrating its structure-preserving properties and validating it through simulations of aggregation-diffusion equations.

Contribution

It introduces a novel particle discretization scheme that maintains the gradient flow structure at the particle level for both linear and nonlinear diffusion.

Findings

The scheme preserves the gradient flow structure.

Simulations confirm the method's validity for aggregation-diffusion equations.

Detailed analysis of one-dimensional cases.

Abstract

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.