# A multi-scale particle method for mean field equations: the general case

## TL;DR

This paper introduces a multi-scale meshfree particle method capable of efficiently approximating mean field equations across different interaction scales, including hyperbolic limits, with applications in swarming, traffic, and granular flow simulations.

## Contribution

It extends previous methods to a more general case, including hyperbolic limits, and demonstrates a numerical transition between microscopic and macroscopic modeling approaches.

## Key findings

- High computational efficiency near the macroscopic limit.
- Effective handling of both large and small interaction radii.
- Versatile application to various mean field problems.

## Abstract

A multi-scale meshfree particle method for macroscopic mean field approximations of generalized interacting particle models is developed and investigated. The method is working in a uniform way for large and small interaction radii. The well resolved case for large interaction radius is treated, as well as underresolved situations with small values of the interaction radius. In the present work we extend the approach from [39] for porous media type limit equations to a more general case, including in particular hyperbolic limits. The method can be viewed as a numerical transition between a DEM-type method for microscopic interacting particle systems and a meshfree particle method for macroscopic equations. We discuss in detail the numerical performance of the scheme for various examples and the potential gain in computation time. The latter is shown to be particularly high for situations near the macroscopic limit. There are various applications of the method to problems involving mean field approximations in swarming, tra?c, pedestrian or granular fow simulation.