A one-parameter family of realizability-interior closures for odd-order kinetic moment systems

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 decomposes into a boundary term, 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 $M\ge 5$. 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 of this decomposition. The weighted AM-GM inequality makes the accuracy-robustness tradeoff precise: the geometric endpoint ($\eta =0$) is 2-4\% more accurate on bimodal benchmarks, while the arithmetic endpoint ($\eta =1$, Morin-McDonald) provides the most robust hyperbolicity profile. 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 at every tested order, and for a source-compatible choice of the symmetrizer weights, the BGK source dissipates it near equilibrium. A smooth nonlinear entropy exists for $M=3$ but does not for $M=5$ or $M=7$ (certified by linear programming). The closure is validated on bimodal and Mott-Smith benchmarks, where the interpolated family achieves 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 $9$.

38 pages, 8 figures, 11 tables