Yuwei Fan

Model Reduction of Kinetic Equations by Operator Projection

By a further study of the mechanism of the hyperbolic regularization of the moment system for the Boltzmann equation proposed in Cai et al. (Commun Math Sci 11(2):547–571, 2013), we point out that the key point is treating the time and space derivative in the same way. Based on this understanding, a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed. The framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment systems by routine calculations. Almost all existing globally hyperbolic moment systems can be included in the framework, as well as some new moment systems including globally hyperbolic regularized versions of Grad’s ordered moment systems and a multi-dimensional extension of the quadrature-based moment system.

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...

13-Moment System with Global Hyperbolicity for Quantum Gas

We point out that the quantum Grad’s 13-moment system (Yano in Physica A 416:231–241, 2014) is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad’s expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new hyperbolic regularization theory developed in Cai et al. (SIAM J Appl Math 75(5):2001–2023, 2015) and Fan et al. (J Stat Phys 162(2):457–486, 2016). This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the hyperbolic regularization theory.

Quantum hydrodynamic model of density functional theory

In this paper, we extend the method in Cai et al. (J Math Phys 53:103503, 2012) to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn–Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn–Sham equation is a non-linear Schrödinger equation, and the corresponding Wigner equation can be derived as an alternative implementation of DFT. We derive quantum hydrodynamic models of the Wigner equation by moment closure following Cai et al. (J Math Phys 53:103503, 2012). The derived quantum hydrodynamic models are globally hyperbolic thus locally wellposed. The contribution of the Kohn–Sham potential is turned into a nonlinear source term of the hyperbolic moment system. This work provides a new possible way to solve DFT problems.