Abstract. 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.

In this paper, we propose a globally hyperbolic regularization to the general Grad's moment system in multidimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment system can be analytically given and depend only on the macroscopic velocity and the temperature. The structure of the eigenvalues and eigenvectors of the coefficient matrix is fully clarified. The regularization together with the properties of the resulting moment systems is consistent with the simple onedimensional case discussed in [1]. In addition, all characteristic waves are proven to be genuinely nonlinear or linearly degenerate, and the studies on the properties of rarefaction waves, contact discontinuities, and shock waves are included. GLOBALLY HYPERBOLIC MOMENT SYSTEM 465 Ã C PROOF. Let i D N D .˛/, with j˛j Ä M . Then we need only to verify that (3.20) z A M .i; 1WN / r D r w i is always valid. Since A M is determined by (3.2), (3.3), and (3.4), and y A M and z A M are defined as in (3.11) and (3.13), respectively, we can write all entries of z A M . Now let us verify equation (3.20) case by case: (1) For˛D 0, z A M .1; N D .e 1 // D 1 is the only nonzero entry of z A M .1; 1WN /; hence, z A M .i; 1WN / r D 1 r u 1 D r D r w i :(2) For˛D e 1 , z A M .i; 1WN / r D 2 r p 2e 1 =2 D 2 r D r u 1 D r w i :(3) For˛D e k , k D 2; : : : ; D, z A M .i; 1WN / r D 1 r p e 1 Ce k =2 D r u k D r w i : ÃHe M 1 . / ZHENNING CAI

We introduce a numerical method for solving Grad's moment equations or regularized moment equations for arbitrary order of moments. In our algorithm, we do not explicitly need the moment equations. Instead, we directly start from the Boltzmann equation and perform Grad's moment method [12] and the regularization technique [27] numerically. We define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions. For the collision term, the BGK model is adopted so that the production step can be done trivially based on the Hermite expansion. Extensive numerical examples for one-and two-dimensional problems are presented. Convergence in moments can be validated by the numerical results for different number of moments.

By a further investigation on the structure of the coefficient matrix of the globally hyperbolic regularized moment equations for Boltzmann equation in [1], we propose a uniform framework to carry out model reduction to general kinetic equations, to achieve certain moment system. With this framework, the underlying reason why the globally hyperbolic regularization in [1] works is revealed. The even fascinating point is, with only routine calculation, existing models are represented and brand new models are discovered. Even if the study is restricted in the scope of the classical Grad's 13-moment system, new model with global hyperbolicity can be deduced.

Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inversepower-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construct new computationally affordable collision models. Numerical experiments show that our method captures the low-order moments very efficiently.