A NEW ALGORITHM FOR BLE (BOLTZMANN LATTICE. EQUATION) NUMERICS. WILLIAM GREENBERG AND MICHAEL WILLIAMS. Abstract. The nonlinear ...
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A NEW ALGORITHM FOR BLE (BOLTZMANN LATTICE EQUATION) NUMERICS WILLIAM GREENBERG AND MICHAEL WILLIAMS Abstract. The nonlinear Boltzmann equation with a discretized spatial variable and finite velocity set (Boltzmann Lattice Equation or BLE) has become a standard tool for numerical studies of particle transport, with applications as diverse as vehicle traffic modeling and E. coli dispersal in humans. The derivation of the Boltzmann equation hinges on diluteness and point particle assumptions. The nonlinear Enskog equation was formulated to take into account nonzero molecular diameters and more dense regimes. We propose here a discretization of the Enskog equation which maintains the non-point particle modeling. This ELE scheme should be an improvement for the discrete modeling of medium and large diameter molecules. Open problems related to ELE are presented.
1. INTRODUCTION Let f (r, v , t) represent the density of particles, of a gas, say, of identical mass particles at position r ∈ Λ with velocity v ∈ S at time t, where Λ ⊂ Rn specifies the physical domain in Rn and the velocity may be restricted to set S ⊂ Rn . We assume f ∈ C([0, ∞), L1 (Λ × S)) If the particles do not interact, and Λ = S = Rn , then the density f satisfies the free streaming equation ∂f (r, v , t) + v · ∇r f (r, v , t) = 0. (1.1) ∂t For initial condition f (r, v , 0) = f0 (r, v ) ∈ L1 (Rn × Rn ), the solution of the Cauchy problem is given by the free streaming semigroup U0 (t): (1.2)
f (r, v , t) = (U (t)f0 )(r, v ) = f0 (r − v t, v )
In 1872, L. Boltzmann introduced a collision operator JB , thereby formulating the most famous equation of kinetic theory: ∂f + v · ∇r f = JB (f, f ) (1.3) ∂t where dv1 d {f f1 − f f1 }B(, v − v1 ) (1.4) JB (f, f ) = R3 ×S 2
To understand the notation, we consider two (identical mass) colliding particles with pre-collision velocities v , v1 producing post-collision velocities v , v1 . Then conservation of momentum and energy implies (1.5)
v = v + ( · (v − v1 )) v1 = v1 − ( · (v − v1 )).
1991 Mathematics Subject Classification. 82C40, 76P05. Key words and phrases. Enskog equation, kinetic theory, Boltzmann lattice, Enskog lattice. 1
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We then write f = f (r, v , t), f1 = f (r, v1 , t), etc. Kernel B(, v − v1 ) enables the Boltzmann equation to model a large class of intermolecular potentials. The Boltzmann equation, which can be derived from the Liouville equation and the BBGKY heirarchy with assumptions implying a dilute gas of point particles, ie., with molecular diameter zero, leads to trivial (ideal gas) transport coefficients. Nevertheless, it has been an outstanding tool in the accurate modeling of gases in the dilute regime for more than a century. In 1922, D. Enskog proposed a modification of the Boltzmann equation to better describe dense gases. The Enskog equation, as revised in the 1960’s to represent exact hydrodynamics, may be written (1.6)
∂f + v · ∇r f = ∂t = dv1 d {Y (r, r + a)f (r)f1 (r + a) − Y (r, r − a)f (r)f1 (r − a)} R3 ×S 2
where Y is function of the local density f (r, v , t) dv , which is to be determined by a Chapman-type procedure. Although the Enskog equation models only hard sphere collisions without intermolecular potential, it has turned out to be an accurate description of dense gases up to ten percent of close packing, To take into account realistic van der Waal’s intermolecular potentials, Greenberg et al. have considered an Enkog type collision operator with square well and, more generally, local piecewise constant potentials.[1, 2]. We wish to build a discrete scheme based on such a generalized Enskog equation. 2. DISCRETIZATION OF THE EQUATIONS Spatial discretization of the Boltzmann equation or the Enskog equation poses these kinetic equations on a lattice Λ with a finite difference approximation for the gradient term, (2.1)
∂fm + (Af )m = J(f, f )m . ∂t
where the velocity variable has been suppressed. The index m is the spatial index denoting the mth lattice point.. To give A specifically (for a periodic threedimensional cubic lattice), let π be an identification between the lattice Λ and a cubic subset of Z 3 . Then A is an N 3 × N 3 matrix: ˆ (2.2) Amn = (v · u ˆ)∆umn (v ) u ˆ
where (2.3)
ˆ (v ) = δmn − δm,π(π−1 (n)+ˆu) , ∆umn
v · u ˆ > 0,
∆uˆ (−v ) = ∆uˆ (v )∗
and the sum is over the three orthogonal coordinate vectors u ˆ. We have, for convenience, taken the lattice spacing to be of unit length. The periodic boundary conditions are imposed by viewing the lattice as a three-dimensional torus, and ˆ ∈ Λ3 for every j. thus π −1 (n) + u Velocity discretization depends upon identifying a set S of finite cardinality which is closed under the conservation laws. We shall label the velocities vj ∈ S for j = 1, . . . , card(S). The distribution function f now should be written as fm,i ,
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where the first index refers to the spatial variable and the second to the velocity variable. Then the Boltzmann lattice equation (BLE) can be written ij ∂fm,i + (Af )m,i = Γkl (fm,k fm,l − fm,i fm,j ). (2.4) ∂t j,k,l
As there is an extensive literature on discrete velocity Boltzmann models going back several decades, we leave to the reader any further study of BLE. On the other hand, the literature on discrete velocity Enskog models is sparse. Here we outline the Enskog model collision operator. If the probability of the collision of two particles of velocities vi and vj producing velocities vk and vl at transfer angle (vector) ˆ is given by Pijkl , then, for a hard sphere collision model, P must satisfy: ) ≤ 1 0 ≤ Pijkl (ˆ
(2.5)
(2.6)
< vi − vj , ˆ > ≥ 0 < v − v , ˆ > ≤ 0 k l Pijkl (ˆ ) = 0 if and only if + v = vk + vl v i j 2 2 vi + vj = vk2 + vl2
(2.7)
kl
Pijkl (ˆ ) =
Pijkl (ˆ ) = 1
ij lk ) = Pji (−ˆ ) Pijkl (ˆ
(2.8)
Then the fully discrete kinetic equation is written ∂fm,i (t) + (Af )m,i (t) = J(f (t), f (t))m,i (2.9) ∂t where ij + J(f, f )m,i = (2.10) Ym,m− Pkl (ˆ )fm,k fm−,l < ˆ, vl − vk > j,k,l
ˆ
− Ym,m+ Pijkl (ˆ )fm,i fm+,j · < ˆ, vi − vj > The geometric factor Ym1 ,m2 is a function of i fm1 ,i , i fm2 ,i and is assumed positive and symmetric in its arguments. An existence theorem for ELE has been presented by Greenberg and Williams under the indicated constraints.[3] In particular, writing L1 (Λ × S) for the less familiar but more sensible notation RΛ×S : Theorem 2.1. For any initial condition f0 ∈ L1 (Λ × S), f0 ≥ 0, the initial value problem for ELE has a unique solution f ∈ C([0, ∞), L1 (Λ × S), and f (t) ≥ 0. 3. OPEN PROBLEMS So little is known about ELE that nearly all questions involving construction of specific models are open. From a physical point of view, however, ELE derives from a kinetic equation which takes explicitly into account the nonzero diameter of colliding particles. Since applications are nearly always either to non-dilute systems or else to systems with molecules of substantial size, there is every reason to believe that a numerical calculation using ELE will have the potential to be substantially more accurate than BLE. Again, however, actual numerical comparisons have, to
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date, not been made. For that reason, this subject appears to be one with great research potential. An example of ELE for a cubic lattice is a generalization of the Carleman model for the Boltzmann equation. In this case, Λ = {0, 1, . . . , N }3 , say, and the discrete velocity set is S = {ˆ x, −ˆ x, yˆ, −ˆ y , zˆ, −ˆ z },
(3.1)
where the velocities in S are vectors parallel to the coordinate axes, normalized for the sake of convenience. A more complicated model on Λ, related to the Broadwell model, is (3.2) S = {±ˆ x, ±ˆ y , ±ˆ z , ±(ˆ x − yˆ), ±(ˆ x − zˆ), ±(ˆ y − zˆ), ±(ˆ x + yˆ), ±(ˆ x + zˆ), ±(ˆ y + zˆ), 0} It is not difficult to construct a similar set of velocities aligned to the directions implicit in a triangular lattice. Using the continuum Enskog equation with square well as model, a version of ELE with nearest neighbor type interactions can be derived, with an extended collision operator where the set of velocities S = S1 ⊕ S2 ⊕ S3 divides into subsets respecting the additional energy of the square well. More precisely, S1 represents collisions at the molecular diameter, S2 entrance to the square well, and S3 exit from the square well. In this case, energy conservation for S2 and S3 ”collisions“ imply: 1 1 (3.3) vk = vi − σ {< , vi − vj > −[< σ , vi − vj >2 +4E] 2 } 2 1 1 (3.4) vl = vj + σ {< σ , vi − vj > −[< σ , vi − vj >2 +4E] 2 } 2 for entrance to the well of depth E, and 1 1 vk = vi − τ {< τ , vi − vj > −[< τ , vi − vj >2 −4E] 2 } (3.5) 2 1 1 (3.6) vl = vj + τ {< τ , vi − vj > −[< τ , vi − vj >2 −4E] 2 } 2 for exit from the well. Then ELE has a collision operator of the form (3.7)
J(f, f )m,i = J(f, f )m,i () + γJ(f, f )m,i (σ ) + γJ(f, f )m,i (τ )
where represents momentum transfer to nearest neighbors, σ , τ represent momentum transfer to a set of next nearest neighbors, and the notation should be transparent. Details are left for the reader (as a generalization of [1, 2]). The authors believe developments in this direction, which have to date not appeared in the literature, will provide a rich set of important research questions. References [1] W. Greenberg, P. Lei and R.S. Liu, Stability theory for the kinetic equations of a moderately dense gas, in Rarefied Gas Dynamics: Theory and Simulations (Progress in Astronaut. and Aeronaut., vol. 159), American Inst. Aeron. Astron., Washington, DC, 1994, pp. 599–607. [2] W. Greenberg and A. Yao, Kinetic equations for gases with piecewise constant intermolecular potentials, Transport Theory and Stat. Phys. 27, 137–150 (1997). [3] W. Greenberg and M. Williams, Global Solutions of the Enskog Lattice Equation with Square Well Potential, preprint. Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061 USA
