Globally hyperbolic moment model of arbitrary order for three-dimensional special relativistic Boltzmann equation

TL;DR

This paper develops a high-order, globally hyperbolic moment model for the 3D special relativistic Boltzmann equation, ensuring stability and Lorentz covariance, advancing the mathematical modeling of relativistic kinetic theory.

Contribution

It introduces a novel arbitrary order moment system based on operator projection, orthogonal polynomials, and spherical harmonics, with proven hyperbolicity and stability.

Findings

The moment system is globally hyperbolic.

The model is linearly stable.

Lorentz covariance is established in 1D space.

Abstract

This paper extends the model reduction method by the operator projection to the three-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order moment system is built on our careful study of infinite families of the complicate Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics. We derive the recurrence relations of the polynomials, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros. The recurrence relations and partial derivatives of the real spherical harmonics are also given. It is proved that our moment system is globally hyperbolic, and linearly stable. Moreover, the Lorentz covariance is also studied in the 1D space.