A Mass Preserving Numerical Scheme for Kinetic Equations that Model Social Phenomena

TL;DR

This paper introduces a new deterministic numerical scheme for kinetic equations modeling social phenomena, which preserves mass, is computationally efficient, and outperforms stochastic agent-based methods in accuracy and resource usage.

Contribution

The paper develops the Mass Preserving Collocation Method, a novel deterministic scheme for kinetic equations with Dirac delta transition rates, ensuring mass conservation and high fidelity simulation.

Findings

The scheme accurately models multiple subsystems in social kinetic equations.

It outperforms stochastic methods like Tau-leaping in efficiency and consistency.

The method requires less computational resources and avoids stochastic variability.

Abstract

In recent years, kinetic equations have been used to model many social phenomena. A key feature of these models is that transition rate kernels involve Dirac delta functions, which capture sudden, discontinuous state changes. Here, we study kinetic equations with transition rates of the form $$ T(x,y,u) = \delta_{\phi(x,y) - u}. $$ We establish the global existence and uniqueness of solutions for these systems and introduce a fully deterministic scheme, the \emph{Mass Preserving Collocation Method}, which enables efficient, high fidelity simulation of models with multiple subsystems. We validate the accuracy, efficiency, and consistency of the solver on models with up to five subsystems, and compare its performance against two state-of-the-art agent-based methods: Tau-leaping and hybrid methods. Our scheme resolves subsystem distributions captured by these stochastic approaches while preserving mass numerically, requiring significantly less computational time and resources, and avoiding variability and hyperparameter tuning characteristic of these methods.