A Positive and Stable L2-minimization Based Moment Method for the Boltzmann Equation of Gas dynamics

TL;DR

This paper introduces a positive, stable L2-minimization based moment method for solving the Boltzmann equation in gas dynamics, ensuring unique, physically meaningful solutions with proven stability and demonstrated accuracy.

Contribution

It proposes a novel L2-minimization approach with positivity constraints for the moment-closure problem, ensuring stability and uniqueness in solutions for the Boltzmann equation.

Findings

The method guarantees positivity of solutions.

A CFL-type condition ensures stability and feasibility.

Numerical experiments confirm accuracy and stability.

Abstract

We consider the method-of-moments approach to solve the Boltzmann equation of rarefied gas dynamics, which results in the following moment-closure problem. Given a set of moments, find the underlying probability density function. The moment-closure problem has infinitely many solutions and requires an additional optimality criterion to single-out a unique solution. Motivated from a discontinuous Galerkin velocity discretization, we consider an optimality criterion based upon L2-minimization. To ensure a positive solution to the moment-closure problem, we enforce positivity constraints on L2-minimization. This results in a quadratic optimization problem with moments and positivity constraints. We show that a (Courant-Friedrichs-Lewy) CFL-type condition ensures both the feasibility of the optimization problem and the L2-stability of the moment approximation. Numerical experiments showcase the accuracy of our moment method.