Modeling of a diffusion with aggregation: rigorous derivation and numerical simulation

TL;DR

This paper rigorously derives and numerically simulates a diffusion-aggregation equation with delta potential, analyzing the transition regimes and validating theoretical results through simulations.

Contribution

It provides a rigorous derivation from stochastic particle systems and compares theoretical results with numerical simulations, especially in challenging regime switch scenarios.

Findings

Successful derivation from stochastic particle systems

Numerical analysis of regime switch where theory fails

Validation of theoretical results through simulations

Abstract

In this paper, a diffusion-aggregation equation with delta potential is introduced. Based on the global existence and uniform estimates of solutions to the diffusion-aggregation equation, we also provide the rigorous derivation from a stochastic particle system while introducing an intermediate particle system with smooth interaction potential. The theoretical results are compared to numerical simulations relying on suitable discretization schemes for the microscopic and macroscopic level. In particular, the regime switch where the analytic theory fails is numerically analyzed very carefully and allows for a better understanding of the equation.