Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries

  title={Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries},
  author={Luc Mieussens},
  journal={Journal of Computational Physics},
  • L. Mieussens
  • Published 10 August 2000
  • Computer Science
  • Journal of Computational Physics
We present new numerical models for computing transitional or rarefied gas flows as described by the Boltzmann-BGK and BGK-ES equations. We first propose a new discrete-velocity model, based on the entropy minimization principle. This model satisfies the conservation laws and the entropy dissipation. Moreover, the problem of conservation and entropy for axisymmetric flows is investigated. We find algebraic relations that must be satisfied by the discretization of the velocity derivative… 
Efficient asymptotic preserving schemes for BGK and ES-BGK models on cartesian grids
This work devise a new wall boundary condition ensuring a smooth transition of the solution from the rarefled regime to the hydrodynamic regime and exploit the ability of Cartesian grids to massive parallel computations (HPC) to drastically reduce the computational time which is an issue for kinetic models.
High-order conservative asymptotic-preserving schemes for modeling rarefied gas dynamical flows with boltzmann-BGK equation
High-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied and potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.
An improved unified gas-kinetic scheme and the study of shock structures
With discretized particle velocity space, a unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the Bhatnagar–Gross–Krook (BGK) model (2010. J. Comput. Phys.,
Conservative numerical methods for advanced model kinetic equations
A direct approach for ensuring conservation in the numerical methods for model kinetic equations, including Shakhov and Rykov models, is presented, based on the approximation of the conditions which are used in deriving these model equations.
Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria
The proposed double-distribution-function based lattice Boltzmann method is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures and opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach.
Lattice Boltzmann models based on the vielbein formalism for the simulation of flows in curvilinear geometries.
This paper considers the Boltzmann equation with respect to orthonormal vielbein fields in conservative form, and derives the macroscopic equations in a covariant tensor notation, and shows that the hydrodynamic limit can be obtained via the Chapman-Enskog expansion in the Bhatnaghar-Gross-Krook approximation for the collision term.
A hybrid method for hydrodynamic-kinetic flow - Part II - Coupling of hydrodynamic and kinetic models


A discrete velocity model of this equation is proposed using the minimum entropy principle to define a discrete equilibrium function, and this model ensures positivity of solutions, conservation of moments, and dissipation of entropy.
Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations
High resolution finite difference schemes for solving the nonlinear model Boltzmann equations are presented for the computations of rarefied gas flows. The discrete ordinate method is first applied
A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics
Abstract We propose a conservative and entropic discrete-velocity method to compute the solutions of the Boltzmann equation in the case of monoatomic species. We begin by defining a discrete
An Implicit Monte Carlo Method for Rarefied Gas Dynamics
For the space homogeneous Boltzmann equation, we formulate a hybrid Monte Carlo method that is robust in the fluid dynamic limit. This method is based on an analytic representation of the solution
Higher Order Approximation Methods for the Boltzmann Equation
A higher order time differencing method for the spatially nonhomogeneous Boltzmann equation is derived from the integral form of the equation along its characteristic line. Similar to the splitting
A method for constructing a model form for the Boltzmann equation
The model of the Boltzmann equation is a useful tool for researching the motion of rarefied gases. The Bhatnagar, Gross, and Krook (BGK) model has been widely used. Gross and Jackson suggested a
Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation
Using the Broadwell model of the nonlinear Boltzmann equation, a second-order scheme is developed that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path.
Convergence of a Weighted Particle Method for Solving the Boltzmann (B.G.K.) Equation
We consider in this paper the numerical solution of the B.G. K. model for the Boltzmann equation. The numerical method used here was introduced by Mas-Gallic (Transport Theory Statist. Phys., 16