# Model reduction of a kinetic swarming model by operator projection

## TL;DR

This paper develops a high-order hyperbolic moment system for the Vicsek swarming model using operator projection, enabling efficient simulation of complex collective behaviors with verified convergence and key feature capturing.

## Contribution

It introduces a novel arbitrary order hyperbolic moment system for the Vicsek model based on operator projection, with proven properties and a semi-implicit numerical scheme.

## Key findings

- The moment system is hyperbolic, rotational invariant, and mass-conservative.
- Solutions converge to the kinetic equation as the order increases.
- The method captures phenomena like shocks, rarefaction, and vortex formation.

## Abstract

This paper derives the arbitrary order globally hyperbolic moment system for a non-linear kinetic description of the Vicsek swarming model by using the operator projection. It is built on our careful study of a family of the complicate Grad type orthogonal functions depending on a parameter (angle of macroscopic velocity). We calculate their derivatives with respect to the independent vari- able, and projection of those derivatives, the product of velocity and basis, and collision term. The moment system is also proved to be hyperbolic, rotational invariant, and mass-conservative. The relationship between Grad type expansions in different parameter is also established. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. It is also compared to the spectral method for the kinetic equation. The results show that the solutions of our hyperbolic moment system converge to the solutions of the kinetic equation for the Vicsek model as the order of the moment system increases, and the moment method can capture key features such as rarefaction and shock waves, contact discontinuity, and vortex formation.