Zhenning Cai

Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation

We introduce a numerical method for solving Grads moment equations or regularized moment equations for an 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 Grads moment method [H. Grad, Commun. Pure Appl. Math., 2 (1949), pp. 331-407] and the regularization technique [H. Struchtrup and M. Torrilhon, Phys. Fluids, 15 (2003), pp. 2668-2680] 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 numbers of moments.

NR

In this paper, we propose a method to simulate the microflows with Shakhov model using the NR

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Simulation of Microflows with Shakhov Model

method developed in [Z. Cai and R. Li, SIAM J. Sci. Comput., 32 (2010), pp. 2875-2907; Z. Cai, R. Li, and Y. Wang, Commun. Comput. Phys., 11 (2012), pp. 1415-1438; Z. Cai, R. Li, and Y. Wang, J. Sci. Comput., to appear]. The equation under consideration is the Boltzmann equation with force terms, and the Shakhov model is adopted to achieve the correct Prandtl number. As the focus of this paper, we derive a uniform framework for different order moment systems on the wall boundary conditions, which is a major difficulty in the moment methods. Numerical examples for both steady and unsteady problems are presented to show the convergence in the number of moments.

A Framework on Moment Model Reduction for Kinetic Equation

Through a deep investigation on the structure of the coefficient matrix of the globally hyperbolic regularized moment equations for the Boltzmann equation in [Z. Cai, Y. Fan, and R. Li, Commun. Math. Sci., 11 (2013), pp. 547--571], we propose a uniform framework for the derivation of reduced models from general kinetic equations. The resulting model appears as a symmetric hyperbolic moment system. This reveals the underlying reason why some models in the literature are hyperbolic while others are not. This framework provides a simple flow chart, following which a number of existing models can be derived in a new way. The framework is also helpful in discovering new models. We apply this to Grads 13-moment distribution function and obtain a new 13-moment model with global hyperbolicity.

Quantum hydrodynamic model by moment closure of Wigner equation

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type first proposed by Grad [“On the kinetic theory of rarefied gases,” Commun. Pure Appl. Math. 2(4), 331–407 (1949)10.1002/cpa.3160020403]. The Grads moment method was originally developed for the Boltzmann equation. Recently, a regularization method for the Grads moment system of the Boltzmann equation was proposed by Cai et al. [Commun. Pure Appl. Math. “Globally hyperbolic regularization of Grads moment system” (in press)] to achieve the global hyperbolicity so that the local well-posedness of the moment system is attained. With the moment expansion of the Wigner function, the drift term in the Wigner equation has exactly the same moment representation as in the Boltzmann equation, thus the regularization applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation turns out to be a linear source term, which can only...

The NRxx method for polyatomic gases

In this paper, we propose a numerical regularized moment method to solve the Boltzmann equation with ES-BGK collision term to simulate polyatomic gas flows. This method is an extension to the polyatomic case of the method proposed in [9], which is abbreviated as the NRxx method in [8]. Based on the form of the Maxwellian, the Laguerre polynomials of the internal energy parameter are used in the series expansion of the distribution function. We develop for polyatomic gases all the essential techniques needed in the NRxx method, including the efficient projection algorithm used in the numerical flux calculation, the regularization based on the Maxwellian iteration and the order of magnitude method, and the linearization of the regularization term for convenient numerical implementation. Meanwhile, the particular integrator in time for the ES-BGK collision term is put forward. The shock tube simulations with Knudsen numbers from 0.05 up to 5 are presented to demonstrate the validity of our method. Moreover, the nitrogen shock structure problem is included in our numerical experiments for Mach numbers from 1.53 to 6.1.

Solving Vlasov Equations Using NRxx Method

In this paper, we propose a moment method to numerically solve the Vlasov equations using the framework of the NR

Numerical Simulation of Microflows Using Moment Methods with Linearized Collision Operator

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Numerical simulation of large hyperbolic moment systems with linear and relaxation production terms

method developed in [Z. Cai and R. Li, SIAM J. Sci. Comput., 32 (2010), pp. 2875--2907; Z. Cai, R. Li, and Y. Wang, J. Sci. Comput., 50 (2012), pp. 103--119] for the Boltzmann equation. Due to the same convection term for the Boltzmann equation and the Vlasov equations, the NR