13-Moment System with Global Hyperbolicity for Quantum Gas

Abstract

We point out that the quantum Grad's 13-moment system [R. Yano, Physica A: Statistical Mechanics and its Applications, 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 theory developed in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015]. 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 theory in [Z. Cai et al., SIAM J. Appl. Math., 75(5):2001-2023, 2015, Y. Fan, J. Stat. Phys., 161(4), 2015].Comment: 23 pages and 12 figure

Direct Flux Gradient Approximation to Close Moment Model for Kinetic Equations

“…Additionally, the moment model by the improved framework does not change the consequence of Maxwellian iteration, which means the NSF laws are preserved. As mentioned above as comparison, direct application of hyperbolic regularization might not preserve this property [13,16]. Furthermore, we show that the HME model [8,9] for the Boltzmann equation, the P N model, the M N model, and the HMP N model for the radiative transfer equation [35,31,16,25] can be regarded as special cases derived by the new approach.…”

“…For instance, in [16], a direct application of the framework changes the M 1 model, which is already globally hyperbolic and based on which the MP N model is derived. On the other hand, in [13,16], it was pointed out that the moment model derived by directly applying the framework in [11,15] lead to incorrect NSF law and Eddington approximation, when a one-step Maxwellian iteration is applied. There has been some recent progress in this direction.…”

Diagram notation for the derivation of hyperbolic moment systems

“…One example of this is the recently developed shallow water moment model, where Legendre polynomials are used to allow for a vertical change of the velocity and similar problems with hyperbolicity occur, see [35]. Further results can be obtained by applying the notation to other moment models, e.g., [17,20,26].…”

“…The only condition is that the ansatz has to be a weighted polynomial in the microscopic velocity variable c, such that a maximum entropy ansatz as in [15,38] is not possible in the current setting. However, many of the moment methods in various applications are based on the expansion with a weighted polynomial basis, see for example [17,23,36,43]. An extension of the diagram notation towards maximum entropy models is an open question and we will focus on weighted polynomial expansions here.…”

“…Numerical simulations [5,12,13,33] demonstrate the efficiency of these regularizations. In recent years, the hyperbolic regularization in [6,21] has been applied to a lot of fields besides gas kinetic theory and microflow, including semiconductor device simulation [10], plasma simulation [14,19], density functional theory [11], quantum gas kinetic theory [17] and rarefied relativistic Boltzmann equation [36]. However, the complexity of the regularized moment models limited their further application and the theoretical and numerical comparison between the regularizations in [6,7] and [30] is not rich.…”