Globally Hyperbolic Regularization of Grad's Moment System

Cai, Zhenning; Fan, Yuwei; Li, Ruo

doi:10.1002/cpa.21472

Cited by 118 publications

(168 citation statements)

References 14 publications

Supporting

Mentioning

163

Contrasting

Unclassified

Order By: Relevance

“…Note that the linearized R13 theory already admits an entropy law, as shown by Struchtrup & Torrilhon (2007). Cai et al (2014) presented a hyperbolic correction to Grad's moment equations that uses a special projection technique. In essence, this approach is considering a distribution function of the form…”

Section: Nonlinear Moment Closuresmentioning

confidence: 96%

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Torrilhon

2016

Annu. Rev. Fluid Mech.

191

158

View full text Add to dashboard Cite

This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number-the ratio between the mean free path of the particles and a macroscopic length scale-becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann's equation in kinetic gas theory. With moment approximations, extended fluid dynamic equations can be derived, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low-Mach number setting for which both evolution equations and boundary conditions are well established. Conversely, nonlinear, high-speed processes require special closures that are still under development. Current approaches are examined, along with the challenge of computing shock wave profiles based on continuum equations. 429

show abstract

Section: Nonlinear Moment Closuresmentioning

confidence: 96%

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Torrilhon

2016

Annu. Rev. Fluid Mech.

191

158

View full text Add to dashboard Cite

show abstract

“…The collision term on the right-hand side evaluated at the grid point j can still be approximated reasonably well by a function of (U j , W G,j ) only since there is no spatial interaction involved. However, after the discretization, the flux term above is going to depend not only on (U , W G ) but also on the basis quantity u j , T j chosen in (14). This motivates us to consider the following approximate moment equation…”

Section: Symmetries and Galilean Invariantmentioning

confidence: 99%

“…The most well-known example is Grad's 13-moment equations. Much effort has gone into making sure that such moment equations are mathematically well-posed and respect the most important physical constraints, see [9,14,15]. However, in both approaches, one has to introduce some uncontrolled truncation in order to obtain a closed system of partial differential equations (PDE).…”

mentioning

confidence: 99%

Uniformly accurate machine learning-based hydrodynamic models for kinetic equations

Han

et al. 2019

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

A new framework is introduced for constructing interpretable and truly reliable reduced models for multi-scale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first through an autoencoder to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active learning procedure is introduced to help ensure that the data set used is representative enough. The reduced system takes the form of the conventional moment systems and works regardless of the numerical discretization used. Numerical results are presented for the BGK model. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow.

show abstract

“…However, it does not provide clues for other models to gain hyperbolicity. The first globally hyperbolic regularization was proposed in [7,8], where the authors study the regularization by investigating the coefficient matrix of the reduced model. This work was extended to a general framework on deriving hyperbolic reduced model for generic kinetic equations in [9,21] based on the operator projection (or truncation).…”

Section: Introductionmentioning

confidence: 99%

A nonlinear moment model for radiative transfer equation in slab geometry

Fan

Zheng

2020

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

Linear models for the radiative transfer equation have been well developed, while nonlinear models are seldom investigated even for slab geometry due to some essential difficulties. We have proposed a moment model in [22] for slab geometry which combines the ideas of the classical PN and MN model. Though the model is far from perfect, it was demonstrated to be quite efficient in numerically approximating the solution of the radiative transfer equation, that we are motivated to further improve this model. Consequently we propose in this paper a new model following the chartmap in [22] with some significant theoretic progresses. The new model is derived with global hyperbolicity, and meanwhile some necessary physical properties are preserved. We give a complete analysis to the characteristic structure and propose a numerical scheme for the new model. Numerical examples are presented to demonstrate the numerical performance of the new model.

show abstract

Globally Hyperbolic Regularization of Grad's Moment System

Cited by 118 publications

References 14 publications

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Modeling Nonequilibrium Gas Flow Based on Moment Equations

Uniformly accurate machine learning-based hydrodynamic models for kinetic equations

A nonlinear moment model for radiative transfer equation in slab geometry

Contact Info

Product

Resources

About