A Blob Method for the Aggregation Equation

TL;DR

This paper introduces a numerical blob method for the aggregation equation that improves convergence rates and preserves key physical properties, offering a robust alternative to traditional particle methods.

Contribution

It develops a novel blob method with higher convergence rates and broader kernel admissibility, extending classical vortex blob techniques to the aggregation equation.

Findings

Achieves arbitrarily high polynomial convergence rates

Conserves mass and decreases energy in the particle system

Validates convergence rates through numerical examples

Abstract

Motivated by classical vortex blob methods for the Euler equations, we develop a numerical blob method for the aggregation equation. This provides a counterpoint to existing literature on particle methods. By regularizing the velocity field with a mollifier or "blob function", the blob method has a faster rate of convergence and allows a wider range of admissible kernels. In fact, we prove arbitrarily high polynomial rates of convergence to classical solutions, depending on the choice of mollifier. The blob method conserves mass and the corresponding particle system is both energy decreasing for a regularized free energy functional and preserves the Wasserstein gradient flow structure. We consider numerical examples that validate our predicted rate of convergence and illustrate qualitative properties of the method.