[Paper Review] Globally Hyperbolic Moment System by Generalized Hermite Expansion

Yuwei Fan, Ruo Li|arXiv (Cornell University)|Jan 19, 2014

Gas Dynamics and Kinetic Theory19 references3 citations

TL;DR

This paper proposes a globally hyperbolic moment system derived via generalized Hermite expansion using the full temperature tensor, overcoming the local hyperbolicity limitation of classical Grad's 13-moment system. By applying a regularization technique inspired by [4], the method ensures global hyperbolicity and local well-posedness for arbitrary-order moment systems, with explicit characterization of eigenvalues and characteristic waves across all dimensions.

ABSTRACT

In a recent paper [Z.-N. Cai, Y.-W. Fan, and R. Li. Tech Report, Institude of Math, Peking Univeristy(2013)], it was revealed that a modified 13-moment system taking intrinsic heat fluxes as variables, instead of the heat fluxes along the coordinate vectors which is adopted in the classical Grad 13-moment system, attains some additional advantages than the classical Grad 13-moment system, particularly including that the equilibrium is turned to be the interior point of its hyperbolicity region. The modified 13-moment system was actually derived from the generalized Hermite expansion of the distribution function, where the anisotropy of Hermite expansion is specified by the full temperature tensor. We extend the method therein in this paper to high order of generalized Hermite expansion to derive arbitrary order moment systems, and proposed a globally hyperbolic regularization to achieve locally well-posedness similar to the method in [Z. Cai, Y. Fan, and R. Li, Comm. Pure Appl. Math.(online)(2013)]. Furthermore, the structure of the eigen-system of the coefficient matrix and all characteristic waves are fully clarified. The obtained systems provide a systematic class of hydrodynamic models as the refined version of Euler equations, which is gradually approaching the Boltzmann equation with increasing order of the expansion.

Motivation & Objective

• To overcome the fundamental limitation of classical Grad’s 13-moment system, which lacks global hyperbolicity and fails to be well-posed even near equilibrium.
• To extend the modified 13-moment system—based on generalized Hermite expansion with full temperature tensor—to arbitrary-order moment systems in any dimension.
• To establish a globally hyperbolic regularization method that ensures local well-posedness for high-order moment systems derived from the Boltzmann equation.
• To fully characterize the eigenstructure and characteristic wave behavior (genuinely nonlinear, linearly degenerate) of the resulting regularized system.

Proposed method

• Derives arbitrary-order moment systems through generalized Hermite expansion of the distribution function, where anisotropy is specified by the full temperature tensor instead of isotropic polynomials.
• Identifies that standard generalized Hermite systems lack global hyperbolicity, particularly due to the equilibrium lying on the boundary of the hyperbolicity region.
• Applies a regularization technique from [4] to the coefficient matrix of the moment system, ensuring global hyperbolicity and local well-posedness.
• Explicitly computes eigenvalues and eigenvectors of the regularized system, proving that all characteristic waves are either genuinely nonlinear or linearly degenerate.
• Constructs prolongation of eigenvectors from diagonal blocks to the full system, rigorously verifying the consistency and correctness of the eigensystem.
• Uses recurrence relations of generalized Hermite polynomials and properties of the temperature tensor to derive and validate the structure of the coefficient matrix and its spectral properties.

Experimental results

Research questions

• RQ1 Can a high-order moment system derived from the Boltzmann equation be globally hyperbolic, even near equilibrium?
• RQ2 Does replacing isotropic Hermite polynomials with generalized Hermite polynomials—parameterized by the full temperature tensor—improve the hyperbolicity region?
• RQ3 Can a systematic regularization method be constructed to ensure global hyperbolicity and local well-posedness for arbitrary-order moment systems?
• RQ4 What is the complete structure of the eigen-system and characteristic wave behavior (e.g., shock, rarefaction, contact waves) in the regularized moment system?

Key findings

• The equilibrium state lies in the interior of the hyperbolicity region for the proposed generalized Hermite-based 13-moment system, resolving a key flaw of classical Grad’s system.
• The proposed regularization ensures global hyperbolicity for arbitrary-order moment systems, enabling local well-posedness even for high-order expansions.
• All characteristic waves in the regularized system are either genuinely nonlinear or linearly degenerate, which supports robust Riemann problem analysis.
• The eigenvalues and eigenvectors of the coefficient matrix are explicitly computed, and the full spectral structure is analytically characterized.
• The construction of the eigenvector prolongation is proven to be consistent and proper, validating the global hyperbolicity of the regularized system.
• The method systematically approaches the Boltzmann equation as the order of expansion increases, providing a refined hydrodynamic model beyond Euler and Navier-Stokes-Fourier equations.