Convergence of a linearly transformed particle method for aggregation equations

TL;DR

This paper analyzes a linearly transformed particle method for aggregation equations, providing convergence estimates for smooth and singular interaction forces in various norms, advancing numerical analysis in this area.

Contribution

It introduces convergence estimates for the particle method applied to aggregation equations with both smooth and singular forces, including measure-valued solutions.

Findings

Convergence in $L^1$ and $L^ty$ norms for smooth forces.

Convergence in bounded Lipschitz distance for measure-valued solutions.

Error estimates for singular forces up to the existence time of solutions.

Abstract

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in $L^1$ and $L^\infty$ norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in $L^1 \cap L^p$ norm.