Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

AbstractGrad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

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Numerical modeling of thermal force in a plasma for test-ion transport simulation based on Monte Carlo Binary Collision Model

Journal of Computational Physics ◽

10.1016/j.jcp.2011.12.037 ◽

2012 ◽

Vol 231 (8) ◽

pp. 3211-3227 ◽

Cited By ~ 12

Author(s):

Yuki Homma ◽

Akiyoshi Hatayama

Keyword(s):

Numerical Modeling ◽

Monte Carlo ◽

Ion Transport ◽

Binary Collision ◽

Collision Model ◽

Transport Simulation ◽

Simulation Based

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Many-body interactions incorporated into a binary collision model of radiation damage in solids

10.1117/12.456265 ◽

2002 ◽

Author(s):

Alexander I. Melker ◽

Sergei N. Romanov

Keyword(s):

Radiation Damage ◽

Binary Collision ◽

Many Body ◽

Collision Model ◽

Many Body Interactions

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CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

International Journal of Modern Physics D ◽

10.1142/s0218271800000037 ◽

2000 ◽

Vol 09 (01) ◽

pp. 13-34 ◽

Cited By ~ 6

Author(s):

GEN YONEDA ◽

HISA-AKI SHINKAI

Keyword(s):

General Relativity ◽

Hyperbolic System ◽

Hyperbolic Systems ◽

Equations Of Motion ◽

Well Posedness ◽

Symmetric Hyperbolic System ◽

The Cauchy Problem ◽

Long Time ◽

Vacuum Spacetime ◽

Gauge Conditions

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

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EVOLUTION OF CENTRAL MOMENTS FOR A GENERAL-RELATIVISTIC BOLTZMANN EQUATION: THE CLOSURE BY ENTROPY MAXIMIZATION

Reviews in Mathematical Physics ◽

10.1142/s0129055x02001223 ◽

2002 ◽

Vol 14 (05) ◽

pp. 469-510 ◽

Cited By ~ 9

Author(s):

ZBIGNIEW BANACH ◽

WIESLAW LARECKI

Keyword(s):

Vector Field ◽

Boltzmann Equation ◽

Velocity Vector ◽

Hyperbolic Systems ◽

Velocity Vector Field ◽

Field Equations ◽

Entropy Maximization ◽

Central Moments ◽

Relativistic Boltzmann Equation ◽

Lorentzian Metric

Beginning from the relativistic Boltzmann equation in a curved space-time, and assuming that there exists a fiducial congruence of timelike world lines with four-velocity vector field u, it is the aim of this paper to present a systematic derivation of a hierarchy of closed systems of moment equations. These systems are found by using the closure by entropy maximization. Our concepts are primarily applied to the formalism of central moments because if an alternative and more familiar theory of covariant moments is taken into account, then the method of maximum entropy is ill-defined in a neighborhood of equilibrium states. The central moments are not covariant in the following sense: two observers looking at the same relativistic gas will, in general, extract two different sets of central moments, not related to each other by a tensorial linear transformation. After a brief review of the formalism of trace-free symmetric spacelike tensors, the differential equations for irreducible central moments are obtained and compared with those of Ellis et al. [Ann. Phys. (NY)150 (1983) 455]. We derive some auxiliary algebraic identities which involve the set of central moments and the corresponding set of Lagrange multipliers; these identities enable us to show that there is an additional balance law interpreted as the equation of balance of entropy. The above results are valid for an arbitrary choice of the Lorentzian metric g and the four-velocity vector field u. Later, the definition of u as in the well-known theory of Arnowitt, Deser, and Misner is proposed in order to construct a hierarchy of symmetric hyperbolic systems of field equations. Also, the Eckart and Landau–Lifshitz definitions of u are discussed. Specifically, it is demonstrated that they lead, in general, to the systems of nonconservative equations.

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Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation

SIAM Journal on Mathematical Analysis ◽

10.1137/15m1012335 ◽

2015 ◽

Vol 47 (6) ◽

pp. 4720-4742 ◽

Cited By ~ 1

Author(s):

Leif Arkeryd ◽

Anne Nouri

Keyword(s):

Cauchy Problem ◽

Boltzmann Equation ◽

Well Posedness ◽

The Cauchy Problem

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A Multi-Relaxation-Time Finite Volume Discrete Boltzmann Method for Viscous Flows

Volume 2: Computational Fluid Dynamics ◽

10.1115/ajkfluids2019-5034 ◽

2019 ◽

Author(s):

Leitao Chen ◽

Hamid Sadat ◽

Laura Schaefer

Keyword(s):

Relaxation Time ◽

Boltzmann Equation ◽

Finite Volume ◽

Dynamic Models ◽

Viscous Flows ◽

Constitutive Law ◽

Collision Model ◽

New Model ◽

Mesh Resolution ◽

Boltzmann Method

Abstract Conventional constitutive law-based fluid dynamic models solve the conservation equations of mass and momentum, while kinetic models, such as the well-known lattice Boltzmann method (LBM), solve the propagation and collision processes of the Boltzmann equation-governed particle distribution function (PDF). Such models can provide an a priori modeling platform on a more fundamental level while easily reconstructing macroscopic variables such as velocity and pressure from the PDF. While the LBM requires a rigid and uniform grid for spatial discretization, another similar unique kinetic model known as the finite volume discrete Boltzmann method (FVDBM) has the ability to solve the discrete Boltzmann equation (DBE) on unstructured grids. The FVDBM can easily and accurately capture curved and more complicated fluid flow boundaries (usually solid boundaries), which cannot be satisfactorily realized in the LBM framework. As a result, the FVDBM preserves the physical advantages of the LBM over the constitutive law-based model approach, but also incorporates a better boundary treatment. However, the FVDBM suffers larger diffusion errors compared to the LBM approach. Building on our previous work, the FVDBM is further developed by integrating the multi-relaxation-time (MRT) collision model into the existing framework. Compared to the existing FVDBM approach that uses the Bhatnagar–Gross–Krook (BGK) collision model, which is also known as the single-relaxation-time (SRT) model, the new model can significantly reduce diffusion error or numerical viscosity, which is essential in the simulation of viscous flows. After testing the new model, the MRT-FVDBM, and the old model, the BGK-FVDBM, on Taylor-Green vortex flow, which can quantify the diffusion error of the applied model, it is found that the MRT-FVDBM can reduce the diffusion error at a faster rate as the mesh resolution increases, which renders the MRT-FVDBM a higher-order model than the BGK-FVDBM. At the highest mesh resolution tested in this paper, the reduction of the diffusion error by the MRT-FVDBM can be up to 30%.

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