On the convergence of the regularized entropy-based moment method for kinetic equations

TL;DR

This paper proves the convergence of a regularized entropy-based moment method for kinetic equations, addressing implementation challenges and providing theoretical and numerical validation of the method's accuracy.

Contribution

It introduces a convergence proof for the regularized method using relative entropy, including convergence rates and numerical validation.

Findings

Convergence of the regularized method to the original as regularization vanishes

Optimal convergence rates demonstrated numerically

Theoretical validation under bounded velocity domain and Lipschitz conditions

Abstract

The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.