Discretization of the velocity space in solution of the Boltzmann equation

We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to be equivalent to truncating the Hermite expansion of the distribution function to the corresponding order. The truncated part of the distribution has no contribution to the moments of low orders and is negligible at small Mach numbers. Higher order approximations to the Boltzmann equation can be achieved by using more velocities in the quadrature. 47.11.+j,05.20.Dd,02.70.-c Typeset using REVTEX 1 The Boltzmann equation is a well accepted mathematical model of a fluid at the microscopic level. It describes the evolution of the single particle distribution function, f(x, ξ, t), in the phase space (x, ξ), where x and ξ are the position and velocity vectors respectively. This description of a fluid is more fundamental than the Navier-Stokes (NS) equations. It has a broader range of application and provides more detailed microscopic information which is critical for the modeling of the underlying physics behind complex fluid behavior. However, direct solution of the full Boltzmann equation is a formidable task due to the high dimensions of the distribution and the complexity in the collision integral. Among the various techniques developed [1], the discrete velocity method was introduced [2] based on the intuitive assumption that the gas particles can be restricted to have only a small number of velocities. The lattice Boltzmann equation (LBE) method formally falls into this category. The development of the LBE method for simulation of fluid dynamics was independent of the continuum Boltzmann equation. The discrete LBE was first written to describe the dynamics of the distribution function in the lattice gas automaton (LGA) [3,4], in which the fluid physics is simulated at the microscopic level by “Boolean” particles moving with discrete velocities on a regular lattice, mimicking the motion of the constituent particles of a fluid. A Bhatnagar-Gross-Krook (BGK) collision model [5] was later adopted in the LBE in place of the complicated collision term [6,7]. In this lattice Boltzmann BGK model, the equilibrium distribution is chosen a posteriori by matching the coefficients in a small velocity (Mach number) expansion so that the correct hydrodynamic equations can be derived using the Chapman-Enskog method. Recently it has been argued [8,9] that the LBE method can be derived from the continuum Boltzmann equation with a BGK collision model. In the new derivations, the Maxwellian distribution is Taylor expanded to second order in the fluid velocity scaled with the sound speed. Abe [9] employed a special functional form for the distribution function so that the macroscopic fluid variables are completely determined by the values of the distribution function at a set of discrete velocities. By noticing that in the Chapman-Enskog calculation, the functional form of the equilibrium distribution function in velocity space is only relevant in the calculation of the low-order moments, and for the Taylor expanded Maxwellian, those moments can be calculated 2 exactly using a Gaussian quadrature, it is concluded that the NS equations can be derived from the Boltzmann equation evaluated on the nodes of the quadrature [8]. On substituting the weights of the corresponding quadrature into the expansion of the Maxwellian, the coefficients of the LBE equilibrium distribution function are recovered. The Boltzmann equation evaluated at the discrete velocities can then be further discretized in x and t in various ways for numerical integration [8]. The LBE models are shown to correspond to solving the discrete Boltzmann equations with a particular finite difference scheme [10]. The recovery of the NS equations from the Boltzmann equation by using a small number of collocation points in velocity space is not accidental. Almost half century ago, Grad [11] introduced a sequence of approximations to the Boltzmann equation by expanding the distribution function in terms of Hermite polynomials in velocity space. The Hermite coefficients are directly related to the macroscopic fluid variables such as density, velocity, internal energy, stress and so on. By keeping Hermite polynomials of up to third order, Grad obtained a system of equations for thirteen moments of the distribution function. This system of equations, known as the “13 moment” approximation, was argued to be a better approximation than the Chapman-Enskog calculation [12,13]. By noticing that the Hermite coefficients for a given function can be estimated using a Hermite quadrature formula, and that this estimation is exact when the function satisfies certain conditions, an important correspondence between the LBE method and the approximation by Hermite polynomial expansion can be immediately identified. In this Letter, we show that by discretizing the Boltzmann-BGK equation at a set of velocities that correspond to the nodes of a Gauss-Hermite quadrature in velocity space, we effectively project and solve the Boltzmann equation in a subspace spanned by the leading Hermite polynomials. The truncated part of the distribution has no contribution to the low-order moments that appear explicitly in the conservation equations. We start from the following Boltzmann-BGK equation: ∂f ∂t + ξ · ∇f = − τ (f − f ), (1) where, τ is a relaxation time, f (0) is the Maxwellian 3 f (0) = ρ ( m 2πkBT )D/2 e − m 2kBT |ξ−u| , (2) where D is the dimension of the space, kB is the Boltzmann constant, and m is the mass of the molecule. The mass density, ρ, macroscopic fluid velocity, u, and the temperature, T , are all functions of x and t. We introduce the dimensionless quantity θ = Tm0/T0m, where T0 is a characteristic temperature and m0 is an unit of the molecular mass. After rescaling the velocities, ξ and u, in units of the constant c0 = √ kBT0/m0, which is the sound speed in a gas consisting of molecules of mass m0 and at temperature T0, the Maxwellian takes the following simple form: f (0) = ρ (2πθ)D/2 e 1 2θ |ξ−u| . (3) The introduction of m0 is for the gas mixtures of components with different molecular masses. For a single component system we can chose m = m0 and have θ = T/T0. If the time and length scales t0 and L are chosen so that L/t0 = c0, the dimensionless Boltzmann-BGK equation will have the same form as Eq. (1) with τ being the dimensionless relaxation time. The mass density, ρ, the dimensionless fluid velocity, u, and the dimensionless internal energy density, ǫ = Dθ/2, are expressed as the velocity moments of the form ∫ fφ(ξ)dξ, with φ = 1, ξ, and ξ respectively: