Convergence Analysis of the Grad's Hermite Approximation to the Boltzmann Equation

TL;DR

This paper proves the $L^2$-convergence and explicit rates of stable Hermite series approximations for linear kinetic equations, addressing boundary condition issues that previously caused instability in Grad's method.

Contribution

It establishes the $L^2$-convergence of stable Hermite approximations for linear kinetic equations and provides explicit convergence rates under regularity assumptions.

Findings

Confirmed convergence rates through numerical experiments

Addressed boundary condition stability issues in Hermite expansions

Provided explicit convergence rates for the approximations

Abstract

In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad's Hermite expansion, which could result in non-converging solutions. For linear kinetic equations, a method for posing stable boundary conditions was recently proposed for (formally) arbitrary order Hermite approximations. In the present work, we study $L^2$-convergence of these stable Hermite approximations, and prove explicit convergence rates under suitable regularity assumptions on the exact solution. We confirm the presented convergence rates through numerical experiments involving the linearised-BGK equation of rarefied gas dynamics.