On the Holway-Weiss Debate: Convergence of the Grad-Moment-Expansion in Kinetic Gas Theory

TL;DR

This paper revisits the debate on the convergence of moment expansions in kinetic gas theory, clarifying the conditions under which these expansions are valid and addressing misconceptions about sub-shock solutions.

Contribution

It clarifies the conditions for convergence of moment expansions and resolves the debate between Holway and Weiss regarding their validity and implications.

Findings

A general convergence restriction for moment expansions exists.

The restriction is not related to sub-shock solutions.

Numerical evidence supports the revised understanding.

Abstract

Moment expansions are used as model reduction technique in kinetic gas theory to approximate the Boltzmann equation. Rarefied gas models based on so-called moment equations became increasingly popular recently. However, in a seminal paper by Holway [Phys. Fluids 7/6, (1965)] a fundamental restriction on the existence of the expansion was used to explain sub-shock behavior of shock profile solutions obtained by moment equations. Later, Weiss [Phys. Fluids 8/6, (1996)] argued that this restriction does not exist. We will revisit and discuss their findings and explain that both arguments have a correct and incorrect part. While a general convergence restriction for moment expansions does exist, it cannot be attributed to sub-shock solutions. We will also discuss the implications of the restriction and give some numerical evidence for our considerations.