Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

Globally hyperbolic regularization of Grad’s moment system in one-dimensional space

Pages: 547 – 571



Zhenning Cai (School of Mathematical Sciences, Peking University, Beijing, China)

Yuwei Fan (School of Mathematical Sciences, Peking University, Beijing, China)

Ruo Li (School of Mathematical Sciences, Peking University, Beijing, China)


In this paper, we present a regularization to the 1D Grad’s moment system to achieve global hyperbolicity. The regularization is based on the observation that the characteristic polynomial of the Jacobian of the flux in Grad’s moment system is independent of the intermediate coefficients in the Hermite expansion. The method does not rely on the form of the collision at all, thus this regularization is applicable to the system without collision terms. Moreover, the proposed approach is proved to be the unique one if only the last moment equation is allowed to be altered to match the condition that the characteristic speeds coincide with the Gauss-Hermite interpolation points. The hyperbolic structure of the regularized system, including the signal speeds, Riemann invariants, and the properties of the characteristic waves including the rarefaction wave, contact discontinuity, and shock are provided in the perfect formations.


Grad’s moment system, regularization, global hyperbolicity, characteristic wave

2010 Mathematics Subject Classification

35L60, 35Q20, 76P05

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