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

TL;DR

This paper develops a high-order globally hyperbolic moment model for the one-dimensional special relativistic Boltzmann equation, providing a mathematically rigorous and numerically verifiable approach to approximate solutions.

Contribution

It introduces a novel arbitrary order hyperbolic moment system based on new orthogonal polynomials, with proven properties and a semi-implicit numerical scheme.

Findings

The hyperbolic moment system converges to the Boltzmann equation solution as order increases.

Eigenvalues and hyperbolicity of the system are rigorously established.

Numerical results confirm the convergence and stability of the proposed method.

Abstract

This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors,hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. 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. The results show that the solutions of our hyperbolic moment system can converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.