Structure preserving schemes for mean-field equations of collective behavior
This work provides structure-preserving numerical methods for simulating large interacting agent systems, which is important for computational physics and biology.
The authors develop numerical schemes for mean-field equations of collective behavior that preserve structural properties like nonnegativity, conservation laws, entropy dissipation, and stationary solutions. The methods achieve second-order accuracy in transient regimes and arbitrary accuracy asymptotically.
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.