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Learning Invariance Preserving Moment Closure Model for boltzmann-BGK equation

Communications in Mathematics and Statistics 2023年第1期11卷 59-101页

作者： Zhengyi Li Bin Dong Yanli Wang School of Mathematical Sciences Peking UniversityBeijingPeople’s Republic of China Beijing International Center for Mathematical Research&Center for Machine Learning Research Peking UniversityBeijingPeople’s Republic of China Beijing Computational Science Research Center BeijingPeople’s Republic of China

As one of the main governing equations in kinetic theory,the Boltzmann equation is widely utilized in aerospace,microscopic flow,etc.Its high-resolution simulation is crucial in these related areas.However,due to the high dimensionality of the Boltzmann equation,high-resolution simulations are often difficult to achieve numerically.The moment method which was first proposed in Grad(Commun Pure Appl Math 2(4):331-407,1949)is among the popular numerical methods to achieve efficient high-resolution simulations.We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation,which effectively reduces the dimensionality of the problem.However,one of themain challenges is that it leads to an unclosed moment system,and closure is needed to obtain a closedmoment system.It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory.Other than the traditional human designs of closures,the machine learning-based approach has attracted much attention lately in Han et al.(Proc Natl Acad Sci USA 116(44):21983-21991,2019)and Huang et al.(J Non-Equilib Thermodyn 46(4):355-370,2021).In this work,we propose a machine learning-based method to derive a moment closure model for the boltzmann-BGK equation.In particular,the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances,i.e.,the Galilean invariance,reflecting invariance,and scaling invariance,inherited from the original boltzmann-BGK equation and playing an important role in the correct simulation of the Boltzmann equation.Numerical simulations on the 1D-1D examples including the smooth and discontinuous initial condition problems,Sod shock tube problem,the shock structure problems,and the 1D-3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

关键词： Boltzmann equation Moment closure Machine learning Neural networks Invariance preserving

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Heat Transfer Problem for the Boltzmann equation in a Channel with Diffusive Boundary Condition

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Chinese Annals of Mathematics,Series B 2022年第6期43卷 1071-1100页

作者： Renjun DUAN Shuangqian LIU Tong YANG Zhu ZHANG Department of Mathematics The Chinese University of Hong KongShatinHong KongChina School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences Central China Normal UniversityWuhan 430079China Department of Applied Mathematics The Hong Kong Polytechnic UniversityHong KongChina.

In this paper,the authors study the 1 D steady boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperaturesθ0andθ1.This problem was studied by Esposito-Lebowitz-Marra(1994,1995)where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition.However,a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary.In the regime where the Knudsen number is reasonably small,the slip phenomenon is significant near the boundary.Thus,they revisit this problem by taking into account the slip boundary conditions.Following the lines of[Coron,F.,Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation,J.Stat.Phys.,54(3-4),1989,829-857],the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points.Then they will establish a uniform L∞estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.

关键词： Boltzmann equation Compressible Navier-Stokes approximation Slip boundary conditions Chapman-Enskog expansion

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MACROSCOPIC REGULARITY FOR THE Boltzmann equation

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Acta Mathematica Scientia 2018年第5期38卷 1549-1566页

作者：黄飞敏王勇 School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China Academy of Mathematics and System Sciences CAS Beijing 100190 China

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of （x, t） in the region R3 × （0, ＋∞）. More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not hap- pen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.

关键词： Boltzmann equation macroscopic regularity compressible Navier-Stokes equations

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High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann equation

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Communications in Computational Physics 2021年第1期29卷 1-56页

作者： Sebastiano Boscarino Seung-Yeon Cho Giovanni Russo Seok-Bae Yun Department of Mathematics and Computer Science University of Catania95125 CataniaItaly Department of Mathematics Sungkyunkwan UniversitySuwon 440-746Republic of Korea

In this paper,we present a conservative semi-Lagrangian finite-difference scheme for the BGK model.Classical semi-Lagrangian finite difference schemes,coupled with an L-stable treatment of the collision term,allow large time steps,for all the range of Knudsen number[17,27,30].Unfortunately,however,such schemes are not conservative.Lack of conservation is analyzed in detail,and two main sources are identified as its cause.First,when using classical continuous Maxwellian,conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points.However,for a small number of grid points in velocity space such error is not negligible,because the parameters of the Maxwellian do not coincide with the discrete moments.Secondly,the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme.As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number.To treat the first problem and ensure machine precision conservation of mass,momentum and energy with a relatively small number of velocity grid points,we replace the continuous Maxwellian with the discrete Maxwellian introduced in[22].The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in[26].In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving(AP)for the underlying Euler limit,as the Knudsen number vanishes.The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

关键词： BGK model Boltzmann equation semi-Lagrangian scheme conservative correction discrete Maxwellian

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CAUCHY PROBLEM FOR LINEARIZED NON-CUTOFF Boltzmann equation WITH DISTRIBUTION INITIAL DATUM

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Acta Mathematica Scientia 2015年第2期35卷 459-476页

作者：李浩光 School of Mathematics and Statistics Wuhan University

In this article, we study the Cauchy problem for the linearized spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. By using the spectral decomposition, we solve the Cauchy problem with initial datum in the sense of distribution, which contains the dual space of a Gelfand-Shilov class. We also prove that this solution belongs to the Gelfand-Shilov space for any positive time.

关键词： Boltzmann equation spectral decomposition Gelfand-Shilov class distribution initial datum

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Convergence ofADistributional Monte Carlo Method for the Boltzmann equation

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Advances in Applied Mathematics and Mechanics 2012年第1期4卷 102-121页

作者： Christopher R.Schrock Aihua W.Wood Department of Mathematics and Statistics Air Force Institute of TechnologyWright-Patterson AFBOH 45433USA

Direct Simulation Monte Carlo(DSMC)methods for the Boltzmann equation employ a point measure approximation to the distribution function,as simulated particles may possess only a single velocity.This representation limits the method to converge only weakly to the solution of the Boltzmann equation.Utilizing kernel density estimation we have developed a stochastic boltzmann solver which possesses strong convergence for bounded and L∞solutions of the Boltzmann equation.This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC.We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods.Toward this end,we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

关键词： Direct simulation monte carlo rarefied gas dynamics Boltzmann equation convergence proof

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THREE WAY DECOMPOSITION FOR THE Boltzmann equation

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Journal of Computational Mathematics 2009年第2期27卷 184-195页

作者： Ilgis Ibragimov Sergej Rjasanow Elegant Mathematics Ltd. Hanauer Muehle 266564 OttweilerGermany Department of Mathematics and Computer Science University of SaarlandSaarbrückenCermany

The initial value problem for the spatially homogeneous Boltzmann equation is considered. A deterministic numerical scheme for this problem is developed by the use of the three way decomposition of the unknown function as well as of the collision integral. On this way, almost linear complexity of the algorithm is achieved. Some numerical examples are presented.

关键词： Boltzmann equation Deterministic scheme Three way decomposition

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Numerical Method for Boltzmann equation with Soroban-Grid CIP Method

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Communications in Computational Physics 2007年第4期2卷 760-782页

作者： Youichi Ogata Hyo-Nam Im Takashi Yabe Department of Mechanical Sciences and Engineering Tokyo Institute of Technology2-12-1 Meguro-kuOokayama 152-8552Japan.

A new numerical scheme to solve the Boltzmann equation in phase space for rarefied gas is described on the basis of the Cubic Interpolated Propagation(CIP)method.The CIP procedure is extended to adaptive unstructured grid system by the Soroban grid.The grid points in velocity space can move dynamically following the spread of velocity space in a spatially localized manner.Such adaptively moving points in velocity space are similar to the particle codes but can provide higher-orderaccurate solutions.Numerical solutions obtained by the Soroban-grid CIP are examined and the validity is discussed.

关键词： CIP Boltzmann equation Soroban grid.

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Gedanken Experiment for Using Boltzmann equation for Relic Graviton Frequencies, in Pre-Planckian Physics and the Independence of Relic Graviton Density from Either Single Repeating Universe Models or Multiverses

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Journal of High Energy Physics, Gravitation and Cosmology 2016年第1期2卷 83-91页

作者： Andrew Walcott Beckwith Physics Department Chongqing University Huxi Campus Chongqing China

We look at what may occur if Boltzmann equations, as presented by Murayama in 2007, Les Houches, are applied to graviton density in a pre-Planckian universe setting. Two restrictions are in order. First of all, we are assuming a graviton mass on the order of 10?62 grams, as if the pre-Planckian regime does not change the nature of Graviton mass, in its low end. Secondly, we are also assuming that a comparatively low temperature regime (far below the Planckian temperature) exists. Finally we are leaving unsaid what may happen if Gravitational waves enter the Planck regime of ultra-high temperature. With those three considerations, we proceed to examine a Graviton density value resulting from perturbation from low to higher temperatures. In the end an ultra- hot Pre big bang cosmology will yield essentially no early universe information transfer crossovers to our present cosmological system. This is not affected by the choice if we have a single repeating universe, or a multiverse. A cold pre inflationary state yields a very different situation. Initial frequencies of Gravitons, though, as outlined may be different in the multiverse case, as opposed to the single repeating universe case. We close with comments as to Bicep 2, and how this document has material as to how to avoid the BICEP 2 disaster. And about choosing between either the possibility of massless Scalar-Tensor Gravity as the correct theory of gravitation or conventional GR.

关键词： Boltzmann equation Stress Energy Tensor Quantum Bounce Infinite Quantum Statistics Heavy Gravity Third Polarization Massless Scalar-Tensor Gravity

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A Relaxation Scheme for Solving the Boltzmann equation Based on the Chapman-Enskog Expansion

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Acta Mathematicae Applicatae Sinica 2002年第1期18卷 37-62页

作者： Shi Jin Lorenzo Pareschi Marshall Slemrod Department of Mathematics University of WisconsinMadisonVan Vleck HallWI 53706USADepartment of MathematicsUniversity of FerraraVia Machiavelli 35 I-44100Italy & Department of MathematicsUniversity of Wisconsin-MadisonVan Vleck HallWI 53706USA.Department of MathematicsUniversity of Wisconsin-MadisonMadisonWI 53715-1149USA.

In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.

关键词： Boltzmann equation Chapman-Enskog expansion Burnett equations relaxation central schemes

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