On the Holway-Weiss debate: Convergence of the Grad-moment-expansion in kinetic gas theory

Moment expansions are used as a model reduction technique in kinetic gas theory to approximate the Boltzmann equation. Rarefied gas models based on so-called moment equations have become increasingly popular recently. However, in a seminal paper by Holway [“Existence of kinetic theory solutions to the shock structure problem,” Phys. Fluids 7(6), 911–913 (1965)], a fundamental restriction on the existence of the expansion was used to explain subshock behavior of shock profile solutions obtained by moment equations. Later, Weiss [“Comments on ‘Existence of kinetic theory solutions to the shock structure problem’ [Phys. Fluids 7, 911 (1964)],” Phys. Fluids 8(6), 1689–1690 (1996)] argued that this restriction does not exist. We will revisit and discuss their findings and explain that both arguments have a correct and an incorrect part. While a general convergence restriction for moment expansions does exist, it cannot be attributed to subshock solutions. We will also discuss the implications of the restriction and give some numerical evidence for our considerations.