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Abstract:Moment closures at odd truncation order present a fundamental difficulty: the standard Gramian closure saturates the realizability boundary, producing only weak hyperbolicity and failing to preserve Maxwellian equilibrium. We show that every odd-order closure for the one-dimensional kinetic equation admits a decomposition into a boundary term, given by the Schur complement of the Hankel moment matrix, and a positive margin above it. An exact polynomial identity connects this margin to the eigenvalues of the flux Jacobian, reducing hyperbolicity to a root-splitting problem. A dimensional argument proves that no margin depending only on density, velocity, and temperature can produce a hyperbolic system for $M \geq 5$. A one-parameter family $C_{\eta,n}$, $\eta \in [0,1]$, built from normalized Schur-complement ratios, reveals that the Morin-McDonald closure is the arithmetic endpoint. The weighted AM-GM inequality orders the family: the geometric endpoint ($\eta = 0$) is 2-4% more accurate on bimodal benchmarks, while the arithmetic endpoint ($\eta = 1$, Morin-McDonald) is the most robust. All members share the same equilibrium Jacobian, whose spectral radius is 13% ($M = 5$) to 29% ($M = 13$) smaller than Grad's closure, allowing larger CFL time steps. A linearized entropy exists for all $M$, and the BGK source dissipates it near equilibrium; a smooth nonlinear entropy exists for $M = 3$ but provably does not for $M \geq 5$. The closure is validated on bimodal and Mott-Smith benchmarks, achieving errors 10-40x smaller than the Gramian or Grad closures, and demonstrated in free-transport Riemann problems at $M = 5, 7, 9, 11$ and BGK Riemann problems at $M = 5$ and $9$.

Submission history

From: Somdeb Bandopadhyay [view email]
[v1] Wed, 24 Jun 2026 17:05:20 UTC (154 KB)