BGK Equation Research Articles

The BGK Equation as the Limit of an N-Particle System

The spatially homogeneous BGK equation is obtained as the limit of a model of a many particle system, similar to Mark Kac’s charicature of the spatially homogeneous Boltzmann equation.

Extended BGK model for diatomic gases

We propose an extension to Boltzmann BGK equation for diatomic gases. The model satisfies the H theorem and accurately recovers the compressible hydrodynamics for a diatomic gas as its macroscopic limit. The collision term is modified to account for the rotational energy and supplemented with an equation representing the evolution of the rotational energy. We demonstrate the accuracy of lattice Boltzmann implementation of the proposed model by calculating the speed of sound in a diatomic medium.

Phase Transitions and Macroscopic Limits in a BGK Model of Body-Attitude Coordination

In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in {mathbb {R}}^3 as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017).

A high-order accurate unstructured spectral difference lattice Boltzmann method for computing inviscid and viscous compressible flows

In the present work, the spectral difference lattice Boltzmann method (SDLBM) is implemented on unstructured meshes for the solution methodology to be capable of accurately simulating the compressible flows over complex geometries. Both the inviscid and viscous compressible flows are computed by applying the unstructured SDLBM. The compressible form of the discrete Boltzmann–BGK equation with the Watari model is considered and the solution of the resulting system of equations is obtained by applying the spectral difference method on arbitrary quadrilateral meshes. The accuracy and robustness of the unstructured SDLBM for simulating the compressible flows are demonstrated by simulating four problems that are steady inviscid supersonic flow past a bump, steady inviscid subsonic flow over the two-element NACA 4412-4415 airfoil with and without the ground effect, steady viscous transonic flow around the NACA 0012 airfoil and unsteady viscous subsonic flow past two side-by-side cylinders. The results obtained by applying the unstructured SDLBM are in good agreement with those of the available high-order accurate Euler/Navier-Stokes solvers and also the experimental data. The present study introduces the unstructured SDLBM as an appropriate inviscid and viscous compressible LBM flow solver for accurately simulating fluid flows over practical problems.

The BGK approximation of kinetic models for traffic

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.

Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles

In this article we study a system of $N$ particles, each of them being defined by the couple of a position (in $\mathbb{R}^d$) and a so-called orientation which is an element of a compact Riemannian manifold. This orientation can be seen as a generalisation of the velocity in Vicsek-type models such as [Degond et al. 2017; Degond, Motsch 2008]. We will assume that the orientation of each particle follows a jump process whereas its position evolves deterministically between two jumps. The law of the jump depends on the position of the particle and the orientations of its neighbours. In the limit $N\to\infty$, we first prove a propagation of chaos result which can be seen as an adaptation of the classical result on McKean-Vlasov systems [Sznitman 1991] to Piecewise Deterministic Markov Processes (PDMP). As in [Jourdain, M\'el\'eard 1998], we then prove that under a proper rescaling with respect to $N$ of the interaction radius between the agents (moderate interaction), the law of the limiting mean-field system satisfies a BGK equation with localised interactions which has been studied as a model of collective behaviour in [Degond et al. 2018]. Finally, in the spatially homogeneous case, we give an alternative approach based on martingale arguments.

Apparent permeability study of rarefied gas transport properties through ultra-tight VORONOI porous media by Discrete Velocity Method

Accurate prediction of the apparent permeability relies on deep understanding of unconventional gas transport mechanism in microscale. To achieve this goal, it is necessary to develop a controllable generation method for low porosity porous media with shale characteristics and explore advanced numerical method to study non-equilibrium phenomena in porous media. Based on the VORONOI tessellation and its topological relationship, a new method for the generation of controllable low porosity porous media is proposed. Two numerical simulation methods are adopted to study gas flow in VORONOI porous media systematically, including Multiple-Relaxation-Time Lattice Boltzmann Method for the intrinsic permeability and Linearized BGK equations via the Discrete Velocity Method for the apparent permeability influenced by the non-equilibrium effect. The results show that the effective porosity is the decisive factor affecting the intrinsic permeability. Porous media with low porosity and high tortuosity are more affected by the rarefied gas effect. The velocity contours show that velocity gradually increases in the throats perpendicular to the pressure gradient direction due to the rarefied throttling effect, which reduces the mass flow rate and apparent permeability significantly. Furthermore, by considering the influences of both porosity and tortuosity, a high-order apparent permeability correction model is proposed to fit porous media with different structures.

Aeroacoustic simulation of broadband sound generated from low-Mach-number flows using a lattice Boltzmann method

The present paper demonstrates the capability of a numerical method based on the lattice Boltzmann method (LBM) with wall-resolved grid to predict the broadband sound generated from the turbulent boundary layer at low Mach numbers. The present method is based on the lattice BGK equation with the D2Q9 and D3Q15 models, and a multi-scale approach using hierarchically refined grids is proposed to efficiently and simultaneously capture the multi-scale phenomena of turbulent eddies near walls and far-field sound waves. Numerical instabilities caused by the lack of grid resolution are suppressed with a fourth-order implicit filtering scheme. This numerical method is discussed in two benchmark problems and an application to the prediction of the broadband sound generated from the turbulent boundary layer. First, the computational accuracy and speed of the LBM scheme are assessed with a pulse-propagating problem. The results indicate that the LBM can achieve accuracy comparable to the fourth-order central scheme with the four-stage Runge-Kutta method for the compressible Navier-Stokes (N-S) equations and compute 12.3 times faster. These findings suggest that the LBM is an efficient computational method for aeroacoustic simulations. Second, the proposed method is validated by simulating the Aeolian tone generated by the flow past a circular cylinder at Reynolds number of 150 and Mach number of 0.2. The present simulation is compared with a compressible N-S simulation using a high-order finite difference scheme in terms of the wave profile and the propagation speed of the tonal sound. This validation result suggests that the present method is available for direct aeroacoustic simulations of low-Mach-number flows. Finally, the capability of the present method to predict the broadband sound is demonstrated by conducting a wall-resolved simulation for the turbulent flow generated by a short separation bubble over an isolated airfoil at Reynolds number of 2.0×105 and Mach number of 0.058. This simulation shows a good agreement with measurements of the surface pressure distributions, the wake velocity profiles, and the far-field sound spectrum. In contrast to hybrid approaches based on the incompressible N-S equations, the present method can accurately predict the broadband sound in the high-frequency range by simulating the acoustic scattering on the airfoil.

Global weak solutions to the relativistic BGK equation

In this paper, the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity, and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.

High order parallelisation of an unstructured grid, discontinuous-Galerkin finite element solver for the Boltzmann–BGK equation

ABSTRACTThis paper outlines the implementation and performance of a parallelisation approach involving partitioning of both physical space and velocity space domains for finite element solution of the Boltzmann-BGK equation. The numerical solver is based on a discontinuous Taylor–Galerkin approach. To the authors' knowledge this is the first time a ‘high order’ parallelisation, or `phase space parallelisation', approach has been attempted in conjunction with a numerical solver of this type. Restrictions on scalability have been overcome with the implementation detailed in this paper. The developed algorithm has major advantages over continuum solvers in applications where strong discontinuities prevail and/or in rarefied flow applications where the Knudsen number is large. Previous work by the authors has outlined the range of applications that this solver is capable of tackling. The paper demonstrates that the high order parallelisation implemented is significantly more effective than previous implementations at exploiting High Performance Computing architectures.

Discontinuous Galerkin discretizations of the Boltzmann–BGK equations for nearly incompressible flows: Semi-analytic time stepping and absorbing boundary layers

We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order conservation law. A stabilized unsplit perfectly matching layer (PML) formulation is introduced for the resulting nonlinear flow equations. The proposed PML equations exponentially absorb the difference between the nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic time discretization methods to improve the time step restrictions in small relaxation times. We also introduce a multirate semi-analytic Adams–Bashforth method which preserves efficiency in stiff regimes. Accuracy and performance of the method are tested using distinct cases including isothermal vortex, flow around square cylinder, and wall mounted square cylinder test cases.

Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations

We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.

Resolving Knudsen layer by high-order moment expansion

We model the Knudsen layer in Kramers’ problem by the linearized high-order hyperbolic moment system. Thanks to the hyperbolicity of the moment system, its boundary conditions are properly reduced from the kinetic boundary condition. For the Kramers’ problem, we present the analytical solutions of the linearized moment systems. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients. With the order of the moment system increasing, the velocity profile approaches to that of the linearized Boltzmann–BGK equation.

A Second-Order Asymptotic-Preserving and Positivity-Preserving Exponential Runge--Kutta Method for a Class of Stiff Kinetic Equations

We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) -- can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving -- can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method.

Numerical boundary conditions in Finite Volume and Discontinuous Galerkin schemes for the simulation of rarefied flows along solid boundaries

We present a numerical comparison between two standard finite volume schemes and a discontinuous Galerkin method applied to the BGK equation of rarefied gas dynamics. We pay particular attention to the numerical boundary conditions in order to preserve the rate of convergence of the method. Most of our analysis relies on a 1D problem (Couette flow), but we also present some results for a 2D aerodynamical flow.

The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium

This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. in [7]. The model is obtained as the diffusive limit of the quantum Liouville–BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium.

The method of modeling the dynamics of a rarefied gas on the basis of the lattice Boltzmann equations and the BGK equation

The method of modeling the dynamics of a rarefied gas on the basis of the lattice Boltzmann equations and the BGK equation

Reduced-order model for the BGK equation based on POD and optimal transport

We present a reduced-order approximation of the BGK equation leading to fast and accurate computations. The BGK model describes the dynamics of a gas flow in both hydrodynamic and rarefied regimes. The particles of the gas are represented by a density distribution function depending on physical space, velocity space and time. In this work, the density distribution function is approximated in the velocity space by a small number of basis functions computed offline. In the offline phase, the BGK equation is sampled in order to collect information on the density distribution function. To complete this sampling, optimal transport is used to add new information by interpolating the samples of the density distribution function. Finally, the basis functions are built by Proper Orthogonal Decomposition. During the online phase, the offline knowledge is used to compute approximations of the density distribution function at low cost. To do so, the BGK equation is projected onto the basis functions, leading to a system of partial differential equations. The system obtained is hyperbolic by construction and is solved by an IMEX Runge–Kutta scheme in time and a finite-volume scheme in space. To improve the accuracy, the reduced-oder model is modified to conserve mass, momentum and energy of the gas. Numerical illustrations of 1D and 2D flows are given. In particular, we investigate the reconstruction and the prediction of shock waves, boundary layers and vortices. The results demonstrate the accuracy of the reduced-order model and the significant reduction of the computational cost.

Gas kinetic scheme for turbulence simulation

The extended gas-kinetic scheme (GKS) for turbulence simulations is developed based on the generalized BGK equation with the effective relaxation time. This relaxation time can be computed from the turbulent viscosity, through which turbulence models can be directly combined. For engineering low-cost simulations of high Reynolds number flows, common-used RANS models are applied, while the LES and hybrid RANS/LES (DES and IDDES) models as well as the minimized dispersion and controllable dissipation (MDCD) reconstruction are adopted in high fidelity turbulence simulations. In addition, the turbulent transport equations are solved in a strongly coupled way by using GKS with scalar transport. The extended GKS is applied in typical turbulent flow predictions including the RANS simulation of hypersonic compression ramp flow and the detailed simulation of multiscale turbulent structures in low-speed cylinder flow with hybrid models. The predicted results agree well with existing experimental measurements and numerical studies, which shows the good accuracy, resolution and robustness of the extended GKS and reveals the wide prospects in turbulence simulations on different model scales, including the multiscale models such as hybrid RANS/LES methods. Furthermore, the multiscale turbulence simulation methods are worthy of further studies based on the generalized BGK model.

Smooth solutions to the BGK equation and the ES-BGK equation with infinite energy

The BGK model and the ES-BGK model of the Boltzmann equation are of great importance in the kinetic theory of rarefied gases. For the Cauchy problems with general initial data, smooth solutions were obtained only in bounded domains for both models. In this paper, we establish the existence and uniqueness of smooth solutions in the whole space. More important, some smooth solutions can permit infiniteness of energy.