Structure preserving schemes for mean-field equations of collective behavior

TL;DR

This paper develops structure-preserving numerical schemes for mean-field equations of collective behavior, ensuring key properties like nonnegativity, conservation, and entropy dissipation, with high accuracy over time.

Contribution

It introduces a generalized Chang-Cooper based method that maintains structural properties and achieves second-order accuracy transiently and arbitrary accuracy asymptotically.

Findings

Schemes preserve nonnegativity and conservation laws.

Methods demonstrate high accuracy in numerical examples.

Approach is general and applicable to various mean-field models.

Abstract

In this paper we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang-Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.