Convergence of a particle method for diffusive gradient flows in one dimension

TL;DR

This paper proves the convergence of a particle-based numerical method for simulating diffusive gradient flows in one dimension, ensuring the method preserves the mathematical structure of the flow.

Contribution

It introduces a particle method that discretizes energy with non-overlapping balls, maintaining the gradient flow structure and providing a rigorous convergence proof.

Findings

The particle method converges to the true diffusive gradient flow.

The energy discretization preserves the gradient flow structure.

The proof uses an abstract convergence result for curves of maximal slope.

Abstract

We prove the convergence of a particle method for the approximation of diffusive gradient flows in one dimension. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles and preserves the gradient flow structure at the particle level. The strategy of the proof is based on an abstract result for the convergence of curves of maximal slope in metric spaces.