h theorem and sufficient conditions for the discrete

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Modern Physics Letters B Vol. 27, No. 1 (2013) 1350007 (15 pages) c World Scientific Publishing Company

DOI: 10.1142/S0217984913500073

H THEOREM AND SUFFICIENT CONDITIONS FOR THE DISCRETE VELOCITY DIRECTION MODEL

Mod. Phys. Lett. B 2013.27. Downloaded from www.worldscientific.com by 58.20.127.106 on 06/13/14. For personal use only.

ZHENYU ZHANG∗ , CHENG PENG and JIANZHONG XU Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China ∗ [email protected] Received 10 October 2012 Revised 24 October 2012 Accepted 25 October 2012 Published 23 November 2012 In this paper, we prove that an H theorem exists for the discrete velocity direction model and provide a set of sufficient conditions, in which the speed distribution functions can be both at equilibrium state and at nonequilibrium state in all flow regimes. To enhance the numerical stability, we simplify the governing equations of the discrete velocity direction model by these sufficient conditions and apply these sufficient conditions to boundary conditions and initial conditions. Keywords: H theorem; discrete velocity direction model; Boltzmann equation; sufficient conditions.

1. Introduction The discrete velocity direction (DVD) model is an approximate method to the Boltzmann equation, which provides an alternative technique to rarefied gas flows.1 The main purpose of the DVD model is to reduce the number of dimensions of Boltzmann equation. In this model, molecules are restricted into a set of fixed directions in the three-dimensional space. These discrete directions are shown in Fig. 1. By this approximation, a gas is described by a set of speed distribution functions and then the Boltzmann equation in the six-dimensional phase space is replaced by a set of equations in the three-dimensional space. The governing equations of DVD model are still differential-integral equations and the integral terms in these governing equations all contain some products of unknown variables. Therefore, the difficulty in mathematics of the Boltzmann equation is not released by the DVD model, while by reducing three momentum dimensions, the computational cost for the Boltzmann equation can be reduced by several orders of magnitude. The DVD model differs from conventional discrete methods to reduce the Boltzmann equation, such as the discrete velocity models (DVM)2,3 and the lattice 1350007-1

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Z.-Y. Zhang, C. Peng & J.-Z. Xu

5

6

y

1

2

x O


z

7

8

3

Fig. 1.

4

Discrete velocity directions in the DVD model.

Boltzmann models (LBM).4 – 7 In the conventional discrete models, both the molecular speed rate and the molecular velocity directions are discrete. Namely, a set of discrete velocities is employed. When the number of discrete velocities is changed, the governing equations should be reconstructed and the code also should be recomposed. While in the DVD model, only the molecular velocity directions are discrete and the molecular speed rate is still continuous. The variables are a set of three-dimensional speed distribution functions, which are still microscope parameters. In numerical calculations, both the speed rates of discrete velocities and the number of discrete velocities can be employed arbitrarily. Peculiarly, the governing equations will remain unchanged when the number of speed rates changes and what we should do is just changing a parameter in the code in numerical calculations. So it is very easy to increase the numerical accuracy by increasing the number of discrete velocities for the DVD model. In addition, the DVD model is efficient. In continuum regime, a few discrete velocities usually give rather accurate results, under which the DVD model is just several times slower than an N-S solver. At large Knudsen number, the computational cost reduces significantly and the numerical accuracy can be improved by employing more discrete velocities.1 The H theorem is an important subject for a kinetic method because it is closely related to the stability. It has attracted much attention to the approximate methods to the Boltzmann equation in recent years. Yong8,9 proved that an H theorem does not exist for the lattice Boltzmann equation with polynomial equilibrium. He provided a set of sufficient conditions under which an LBE model does not admit an H theorem and given a suggestion to improve the stability of LBM. Chen10 pointed out that the fundamental origin of the instability in thermal lattice 1350007-2

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H Theorem and Sufficient Conditions for the DVD Model

Boltzmann models can be attributed to the violation of a global H theorem. Wagner11 pointed out that there are no convex entropy functions whose local equilibria are the polynomials of the form used in the lattice BGK, so the lattice Boltzmann method is not capable of reproducing the full properties of the continuum Boltzmann equations. Succi12 summarized the main achievements about the H theorem for the lattice Boltzmann method and analyzed the role played by the H theorem in numerical calculations systematically. Compared with the Boltzmann equation, most simplified methods become unstable. The DVD model is still a kinetic method and a new molecular collision operator is introduced, so it immediately leads to the question whether an H theorem exists for the DVD model. In this paper, we shall first prove that an H theorem exists for the DVD model and provide a set of sufficient conditions. This work shows the intrinsic stability of the DVD model. To enhance the numerical stability, we shall apply these sufficient conditions into the boundary conditions and initial conditions. 2. H Theorem and Sufficient Conditions A local H theorem means that the H function at a point in space always decreases with time until the gas reaches the equilibrium state. While a global H theorem means that the total value of H function throughout the whole space will decrease with time. Therefore, the global H function can be regarded as the integral of local H function over the whole space, but the influence by the convective terms should be included. In this section, we first investigate the local H theorem for the DVD model and then discuss the global H theorem later. Consider a spatially homogeneous gas, then the governing equation set for the DVD model can be expressed as the following: Z ∞ ∂f1 1 2 = Cπd (c + c′ )(2f2 f7′ + 2f3 f6′ + 2f5 f4′ − 3f1 f8′ − f7 f2′ − f6 f3′ − f4 f5′ )dc′ , ∂t 3 0 Z ∞ ∂f2 1 = Cπd2 (c + c′ )(2f1 f8′ + 2f6 f3′ + 2f4 f5′ − 3f2 f7′ − f8 f1′ − f3 f6′ − f5 f4′ )dc′ , ∂t 3 0 Z ∞ ∂f3 1 = Cπd2 (c + c′ )(2f1 f8′ + 2f7 f2′ + 2f4 f5′ − 3f3 f6′ − f8 f1′ − f5 f4′ − f2 f7′ )dc′ , ∂t 3 0 Z ∞ ∂f4 1 = Cπd2 (c + c′ )(2f2 f7′ + 2f3 f6′ + 2f8 f1′ − 3f4 f5′ − f7 f2′ − f6 f3′ − f1 f8′ )dc′ , ∂t 3 0 Z ∞ ∂f5 1 = Cπd2 (c + c′ )(2f1 f8′ + 2f6 f3′ + 2f7 f2′ − 3f5 f4′ − f8 f1′ − f3 f6′ − f2 f7′ )dc′ , ∂t 3 0 Z ∞ ∂f6 1 = Cπd2 (c + c′ )(2f2 f7′ + 2f5 f4′ + 2f8 f1′ − 3f6 f3′ − f7 f2′ − f1 f8′ − f4 f5′ )dc′ , ∂t 3 0 Z ∞ ∂f7 1 = Cπd2 (c + c′ )(2f8 f1′ + 2f3 f6′ + 2f5 f4′ − 3f7 f2′ − f1 f8′ − f6 f3′ − f4 f5′ )dc′ , ∂t 3 0 1350007-3

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∂f8 1 = Cπd2 ∂t 3

Z

∞

(c + c′ )(2f7 f2′ + 2f6 f3′ + 2f4 f5′ − 3f8 f1′ − f2 f7′ − f3 f6′ − f5 f4′ )dc′ ,

0


(1) where fi = fi (c, ~r, t) and fi′ = fi (c′ , ~r, t) are speed distribution functions in direction i with molecular speed rate c and c′ , respectively, C is the multi-body collision coefficient, which is defined as the ratio of multi-body collisions to total molecular collisions between two opposite directions and d is the molecular diameter. We define the H function for the DVD model as the sum of integrals in eight discrete directions, then 8 Z ∞ X H= fi ln fi dci . (2) i=1

0

Take its time derivative and substitute the governing equation set, then we gain Z ∞Z ∞ 1 ∂H (c + c′ )(1 + ln f1 )(2f2 f7′ + 2f3 f6′ + 2f5 f4′ − 3f1 f8′ = Cπd2 ∂t 3 0 0 − f7 f2′ − f6 f3′ − f4 f5′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f2 )(2f1 f8′ + 2f4 f5′ + 2f6 f3′ − 3f2 f7′ 3 0 0 − f8 f1′ − f5 f4′ − f3 f6′ )dcdc′ Z ∞Z ∞ 1 2 Cπd (c + c′ )(1 + ln f3 )(2f1 f8′ + 2f4 f5′ + 2f7 f2′ − 3f3 f6′ 3 0 0 − f8 f1′ − f5 f4′ − f2 f7′ )dcdc′ Z ∞Z ∞ 1 2 Cπd (c + c′ )(1 + ln f4 )(2f2 f7′ + 2f3 f6′ + 2f8 f1′ − 3f4 f5′ 3 0 0 − f7 f2′ − f6 f3′ − f1 f8′ )dcdc′ Z ∞Z ∞ 1 2 Cπd (c + c′ )(1 + ln f5 )(2f1 f8′ + 2f6 f3′ + 2f7 f2′ − 3f5 f4′ 3 0 0 − f8 f1′ − f3 f6′ − f2 f7′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f6 )(2f2 f7′ + 2f8 f1′ + 2f5 f4′ − 3f6 f3′ 3 0 0 −f7 f2′ − f1 f8′ − f4 f5′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f7 )(2f3 f6′ + 2f8 f1′ + 2f5 f4′ − 3f7 f2′ 3 0 0 − f1 f8′ − f6 f3′ − f4 f5′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f8 )(2f4 f5′ + 2f6 f3′ + 2f7 f2′ − 3f8 f1′ 3 0 0 − f5 f4′ − f3 f6′ − f2 f7′ )dcdc′ . 1350007-4

(3)

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Equation (3) consists of eight double-integrals, in which the value of each integral will remain unchanged when the molecular speeds c and c′ are exchanged. Take the first integral for instance, we have Z

0

∞Z ∞

(c + c′ )(1 + ln f1 )(2f2 f7′ + 2f3 f6′ + 2f5 f4′ − 3f1 f8′ −f7 f2′ −f6 f3′ −f4 f5′ )dcdc′

0

=

Z

0

∞

Z

∞

(c′ + c)(1 + ln f1′ )(2f7 f2′ + 2f6 f3′ + 2f4 f5′ − 3f8 f1′ − f2 f7′

0


− f3 f6′ − f5 f4′ )dcdc′ .

(4)

Now we exchange the molecular speeds c and c′ in the 1st, 2st, 3st and 5st integrals, while keep the molecular speeds unchanged in other four integrals, then ∂H 1 = Cπd2 ∂t 3

Z

0

∞Z ∞

(c + c′ )(1 + ln f1′ )(2f7 f2′ + 2f6 f3′ + 2f4 f5′ − 3f8 f1′

0

− f2 f7′ − f3 f6′ − f5 f4′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f2′ )(2f8 f1′ + 2f5 f4′ + 2f3 f6′ − 3f7 f2′ 3 0 0 − f1 f8′ − f4 f5′ − f6 f3′ )dcdc′ Z ∞Z ∞ 1 2 Cπd (c + c′ )(1 + ln f3′ )(2f8 f1′ + 2f5 f4′ + 2f2 f7′ − 3f6 f3′ 3 0 0 − f1 f8′ − f4 f5′ − f7 f2′ )dcdc′ Z ∞Z ∞ 1 2 Cπd (c + c′ )(1 + ln f4 )(2f2 f7′ + 2f3 f6′ + 2f8 f1′ − 3f4 f5′ 3 0 0 − f7 f2′ − f6 f3′ − f1 f8′ )dcdc′ Z ∞Z ∞ 1 2 Cπd (c + c′ )(1 + ln f5′ )(2f8 f1′ + 2f3 f6′ + 2f2 f7′ − 3f4 f5′ 3 0 0 − f1 f8′ − f6 f3′ − f7 f2′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f6 )(2f2 f7′ + 2f8 f1′ + 2f5 f4′ − 3f6 f3′ 3 0 0 − f7 f2′ − f1 f8′ − f4 f5′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f7 )(2f3 f6′ + 2f8 f1′ + 2f5 f4′ − 3f7 f2′ 3 0 0 − f1 f8′ − f6 f3′ − f4 f5′ )dcdc′ Z ∞Z ∞ 1 Cπd2 (c + c′ )(1 + ln f8 )(2f4 f5′ + 2f6 f3′ + 2f7 f2′ − 3f8 f1′ 3 0 0 − f5 f4′ − f3 f6′ − f2 f7′ )dcdc′ . 1350007-5

(5)

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Combine these integrals, then the first derivative can be written as Z ∞Z ∞ ∂H 1 = Cπd2 (c + c′ )Adcdc′ , ∂t 3 0 0

(6)

in which A = (1 + ln f1′ )(2f7 f2′ + 2f6 f3′ + 2f4 f5′ − 3f8 f1′ − f2 f7′ − f3 f6′ − f5 f4′ ) + (1 + ln f2′ )(2f8 f1′ + 2f5 f4′ + 2f3 f6′ − 3f7 f2′ − f1 f8′ − f4 f5′ − f6 f3′ ) + (1 + ln f3′ )(2f8 f1′ + 2f5 f4′ + 2f2 f7′ − 3f6 f3′ − f1 f8′ − f4 f5′ − f7 f2′ )


+ (1 + ln f4 )(2f2 f7′ + 2f3 f6′ + 2f8 f1′ − 3f4 f5′ − f7 f2′ − f6 f3′ − f1 f8′ ) + (1 + ln f5′ )(2f8 f1′ + 2f3 f6′ + 2f2 f7′ − 3f4 f5′ − f1 f8′ − f6 f3′ − f7 f2′ ) + (1 + ln f6 )(2f2 f7′ + 2f8 f1′ + 2f5 f4′ − 3f6 f3′ − f7 f2′ − f1 f8′ − f4 f5′ ) + (1 + ln f7 )(2f3 f6′ + 2f8 f1′ + 2f5 f4′ − 3f7 f2′ − f1 f8′ − f6 f3′ − f4 f5′ ) + (1 + ln f8 )(2f4 f5′ + 2f6 f3′ + 2f7 f2′ − 3f8 f1′ − f5 f4′ − f3 f6′ − f2 f7′ ) .

(7)

In Eq. (6), the parameters C, π, d2 and c are all positive, so the sign of the first derivative is only determined by term A. Equation (7) contains many products of speed distribution functions and its sign cannot be determined in the current form. Next, divide term A into two parts and let A1 consist of the first four terms in each line and A2 consist of the first three terms and the last three terms in each line in Eq. (7). Then term A can be expressed as A = A1 + A2 , in which A1 = (1 + ln f1′ )(f7 f2′ − f8 f1′ + f6 f3′ − f8 f1′ + f4 f5′ − f8 f1′ ) + (1 + ln f2′ )(f8 f1′ − f7 f2′ + f5 f4′ − f7 f2′ + f3 f6′ − f7 f2′ ) + (1 + ln f3′ )(f8 f1′ − f6 f3′ + f5 f4′ − f6 f3′ + f2 f7′ − f6 f3′ ) + (1 + ln f4 )(f2 f7′ − f4 f5′ + f3 f6′ − f4 f5′ + f8 f1′ − f4 f5′ ) + (1 + ln f5′ )(f8 f1′ − f4 f5′ + f3 f6′ − f4 f5′ + f2 f7′ − f4 f5′ ) + (1 + ln f6 )(f2 f7′ − f6 f3′ + f8 f1′ − f6 f3′ + f5 f4′ − f6 f3′ ) + (1 + ln f7 )(f3 f6′ − f7 f2′ + f8 f1′ − f7 f2′ + f5 f4′ − f7 f2′ ) + (1 + ln f8 )(f4 f5′ − f8 f1′ + f6 f3′ − f8 f1′ + f7 f2′ − f8 f1′ ) and A2 = (1 + ln f1′ )(f7 f2′ + f6 f3′ + f4 f5′ − f2 f7′ − f3 f6′ − f5 f4′ ) + (1 + ln f2′ )(f8 f1′ + f5 f4′ + f3 f6′ − f1 f8′ − f4 f5′ − f6 f3′ ) + (1 + ln f3′ )(f8 f1′ + f5 f4′ + f2 f7′ − f1 f8′ − f4 f5′ − f7 f2′ ) + (1 + ln f4 )(f2 f7′ + f3 f6′ + f8 f1′ − f7 f2′ − f6 f3′ − f1 f8′ ) + (1 + ln f5′ )(f8 f1′ + f3 f6′ + f2 f7′ − f1 f8′ − f6 f3′ − f7 f2′ ) 1350007-6

(8)

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+ (1 + ln f6 )(f2 f7′ + f8 f1′ + f5 f4′ − f7 f2′ − f1 f8′ − f4 f5′ ) + (1 + ln f7 )(f3 f6′ + f8 f1′ + f5 f4′ − f6 f3′ − f1 f8′ − f4 f5′ ) +(1 + ln f8 )(f4 f5′ + f6 f3′ + f7 f2′ − f5 f4′ − f3 f6′ − f2 f7′ ) .

(9)

Then term A1 can be combined into the following form: A1 = (f7 f2′ − f8 f1′ ) ln

f8 f1′ f8 f1′ f8 f1′ ′ ′ ′ ′ + (f f − f f ) ln + (f f − f f ) ln 6 8 4 8 3 1 5 1 f7 f2′ f1 f3′ f4 f5′


+ (f5 f4′ − f7 f2′ ) ln

f7 f2′ f7 f2′ f6 f3′ + (f3 f6′ − f7 f2′ ) ln + (f5 f4′ − f6 f3′ ) ln . ′ ′ f5 f4 f3 f6 f5 f4′ (10)

Equation (10) contains some terms that in the form (a − b) ln(b/a), so this part is always less than or equal to zero. Next, consider term A2 . It cannot be merged directly in the current form like term A1 . So we rewrite it into the following form A2 = (A2 × 2)/2, finally A2 can be written as A2 =

1 f8 f1′ f8 f1′ f8 f1′ [(f7 f2′ − f3 f6′ ) ln + (f7 f2′ − f5 f4′ ) ln + (f6 f3′ − f2 f7′ ) ln ′ ′ 2 f4 f5 f6 f3 f4 f5′ + (f6 f3′ − f5 f4′ ) ln

f8 f1′ f8 f1′ f8 f1′ ′ ′ ′ ′ + (f f − f f ) ln + (f f − f f ) ln 4 3 4 2 5 6 5 7 f7 f2′ f7 f2′ f6 f3′

+ (f8 f1′ − f4 f5′ ) ln

f7 f2′ f6 f3′ f7 f2′ + (f8 f1′ − f4 f5′ ) ln + (f8 f1′ − f6 f3′ ) ln ′ ′ f3 f6 f2 f7 f5 f4′

+ (f8 f1′ − f6 f3′ ) ln

f6 f3′ f4 f5′ f4 f5′ + (f8 f1′ − f7 f2′ ) ln + (f8 f1′ − f7 f2′ ) ln ] . (11) ′ ′ f2 f7 f5 f4 f3 f6′

Each term in Eq. (11) consists of the molecules in all discrete directions, which is in the form (a − b) ln(c/d), so its sign is not fixed. According to above analysis, we know that the generation rate of H will be negative when (A2 ≤ 0) or (A2 > 0 and A1 + A2 ≤ 0) .

(12)

The inequality (12) is the sufficient and necessary condition under which the DVD model satisfies an H theorem. We know that the entropy production in continuous-time kinetic methods is different from that in discrete-time kinetic methods.12 In continuous-time kinetic methods, such as the Boltzmann equation and the discrete velocity models, the entropy is passively transported along collision-free trajectories, so the H theorem is unaffected by particle advection. Therefore, the local H theorem automatically implies the global H theorem in continuous-time kinetic methods. While in discretetime kinetic methods, such as LBM, both time and space are discrete and the local H function involves the transition probabilities from pre-collisional states to postcollisional states, so the H theorem is affected by the particles advection directly, 1350007-7

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which means that the local H theorem does not lead to the global H theorem. The DVD model is still a continuous-time kinetic method. Though the molecular velocity directions are discrete, the molecular speed distribution functions are still continuous both in space and in time and entropy is still passively transported along collision-free trajectories. Therefore, the local H theorem implies the global H theorem for the DVD model. Consider a simple case A2 = 0. Let each term in the form (a − b) ln(c/d) in Eq. (11) is equal to zero, then we have 24 equations in the form “a = b” and “c = d”, ignore the same equations, then we gain the following equation set: f8 f1′ = f5 f4′ ,

f7 f2′ = f3 f6′ ,

f8 f1′ = f4 f5′ ,

f6 f3′ = f2 f7′ ,

f8 f1′ = f3 f6′ ,

f7 f2′ = f5 f4′ ,

f8 f1′ = f6 f3′ ,

f4 f6′ = f2 f7′ ,

f8 f1′ = f7 f2′ ,

f6 f3′ = f5 f4′ ,

f8 f1′ = f7 f2′ ,

f4 f5′ = f3 f6′ .

(13)

Combine these equations, then we gain f8 f1′ = f1 f8′ = f2 f7′ = f7 f2′ = f3 f6′ = f6 f3′ = f4 f5′ = f5 f4′ .

(14)

To be concise, Eq. (14) can be written as ′ fi f9−i = fj′ f9−j ,

(i, j = 1, 2, 3, 4) .

(15)

If we substitute Eq. (15) in Eq. (3), we will have ∂H/∂t ≤ 0. Therefore, Eq. (15) is a set of sufficient conditions for the DVD model. Broaden the range of the sufficient conditions and let a collision frequency only be equal to that in its opposite direction, then we have ′ fi f9−i = fi′ f9−i ,

(i = 1, 2, 3, 4) .

(16)

Substitute Eq. (16) in Eq. (3), we also have ∂H/∂t ≤ 0, namely Eq. (16) is still a set of sufficient conditions. It means that the DVD model will satisfy an H theorem when two kinds of collisions in a diagonal line have the same collision frequency: (a) the collisions between the molecules with speeds ci and c′9−i ; (b) the collisions between the molecules with speeds c′i and c9−i . In Eq. (16), the speed distribution functions are arbitrary, which can be both at equilibrium state and at nonequilibrium state. In addition, these conditions have nothing to do with the Knudsen number, which denotes that the H theorem exists in all flow regimes.

3. Numerical Stability The governing equations of DVD model are still differential-integral equations, which should be solved numerically. In this section, we shall discuss some issues pertinent to the numerical stability. 1350007-8

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Substitute the sufficient conditions (16) into the original governing equation1 set for the DVD model, then we have


∂f1 ∂f1 ∂f8 → ∂f8 − + c~l1 = + c l8 ∂t ∂~r ∂t ∂~r Z ∞ 1 (c + c′ )(f2 f7′ + f3 f6′ + f5 f4′ − 3f1 f8′ )dc′ , = Cπd2 3 0 ∂f2 ∂f2 ∂f7 ∂f7 + c~l2 = + c~l7 ∂t ∂~r ∂t ∂~r Z ∞ 1 (c + c′ )(f1 f8′ + f6 f3′ + f4 f5′ − 3f2 f7′ )dc′ , = Cπd2 3 0

(17)

∂f3 ∂f3 ∂f6 ∂f6 + c~l3 = + c~l6 ∂t ∂~r ∂t ∂~r Z ∞ 1 = Cπd2 (c + c′ )(f1 f8′ + f7 f2′ + f4 f5′ − 3f3 f6′ )dc′ , 3 0 ∂f4 ∂f4 ∂f5 ∂f5 + c~l4 = + c~l5 ∂t ∂~r ∂t ∂~r Z ∞ 1 = Cπd2 (c + c′ )(f2 f7′ + f3 f6′ + f8 f1′ − 3f4 f5′ )dc′ . 3 0 The present equation set always satisfies a global H theorem, in which both the number of independent equations and the number of collision terms are reduced by the sufficient conditions. In fact, Eq. (17) is not necessary in numerical calculations because the DVD model still may satisfy an H theorem if Eq. (16) does not exist. The properties of the collision terms are also changed in Eq. (17). To illustrate the difference, let us consider the molecules in direction i. In Eq. (1), the collisions between the molecular pairs i ↔ (9 − i) decrease the number of molecules in direction i, while the collisions between the molecular pairs in other directions both decrease and increase the number of molecules in direction i simultaneously. While in Eq. (17), the collisions involved in term A2 cancel out and the collisions between the molecular pairs in other directions only increase the number of molecules in direction i. Now we know that the molecular collisions in term A2 cause the instability for the DVD model. Thus, to improve the stability, this kind of molecular collisions in term A2 should be avoided when we construct a new collision operator including more discrete directions. Next, to enhance the numerical stability for the DVD model, we apply the sufficient conditions (16) into the boundary conditions and the initial conditions. Put the speed distribution functions with the same molecular speed on the same side, then Eq. (16) can be expressed as the form of ratio of speed distribution functions fi f9−i

=

fi′ = Fi , ′ f9−i

(i = 1, 2, 3, 4) ,

1350007-9

(18)

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in which Fi are functions of ~r and t. Narrow the range of Eq. (18) and let Fi be a set of positive real numbers αi , then


fi = αi f9−i ,

(i = 1, 2, 3, 4) .

(19)

Equation (19) is also a set of sufficient conditions, which means that a speed distribution function should be proportional to that in the opposite direction. In the DVD model, the molecular speed rate is still continuous and they should be further cut off into a set of discrete velocities in numerical calculations. The discrete velocities can be derived by the following method: Divide the continuous speed rate into some intervals and choose one speed rate as a discrete speed rate in each interval. For a speed intervals from ca to cb , the discrete speed rate cdis is derived by the following equation Z cb Z cb cdis f dc = f cdc . (20) ca

ca

Assume that ni,j is the molecular number density in direction i with the discrete speed rate cj , then the sufficient conditions (19) should be expressed as ni,j = αi n9−i,j ,

(i = 1, 2, 3, 4) ,

(21)

Equation (21) can be applied in boundary conditions and initial conditions directly in numerical calculations. 4. Conclusion We have proven that an H theorem exists for the DVD model and found a set of sufficient condition in this paper. These sufficient conditions show that the H theorem exists both at equilibrium state and at nonequilibrium state in all flow regimes. To enhance the numerical stability, we have proposed a method to set the boundary conditions and the initial conditions in numerical calculations and simplified the governing equation set. The instability for the DVD model was found by investigating the nature of collision terms. This work is helpful for constructing new collision operators for the DVD model including more discrete direction. Appendix. The Governing Equation Set for the DVD Model The DVD model is a kind of discrete methods to reduce the Boltzmann equation. Different from the conventional discrete methods, such as the lattice Boltzmann method and the discrete velocity models, only the directions of molecules are discrete in the DVD model, while the speed of molecules is still continuous. The variables for the DVD model are a set of speed distribution functions in eight fixed directions, which are shown in Fig. 1. These speed distribution functions are defined as fi (ci , ~r, t), (i = 1, 2, . . . , 8), in which ci (0 ≤ ci < ∞) is the molecular speed rate in direction i, ~r is the position vector and t is the time variable. The meaning of these variables is that in the moment t, c ∈ (ci , ci + dci ) and ~r ∈ (~r, ~r + d~r), the 1350007-10

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number of molecules moving toward direction i is fi (ci , ~r, t)d~rdci . By this approximation, the Boltzmann equation in the six-dimensional phase space can be replaced by eight differential-integral equations in the three-dimensional space. The governing equation set for the DVD model can be expressed as:


Direction 1 ∂f1 (ca , ~r, t) ∂f1 (ca , ~r, t) + ca~l1 ∂t ∂~r Z ∞ 2 = Cπd2 (ca + cb )f7 (cb , ~r, t)f2 (ca , ~r, t)dcb 3 0 Z 2 ∞ (ca + cb )f6 (cb , ~r, t)f3 (ca , ~r, t)dcb + 3 0 Z 2 ∞ + (ca + cb )f4 (cb , ~r, t)f5 (ca , ~r, t)dcb 3 0 Z ∞ − (ca + cb )f8 (cb , ~r, t)f1 (ca , ~r, t)dcb 0

− − −

1 3

Z

∞

1 3

Z

∞

1 3

Z

∞

(ca + cb )f2 (cb , ~r, t)f7 (ca , ~r, t)dcb

0


0

0

(ca + cb )f5 (cb , ~r, t)f4 (ca , ~r, t)dcb .

Direction 2 ∂f2 (ca , ~r, t) ∂f2 (ca , ~r, t) + ca~l2 ∂t ∂~r Z ∞ 2 = Cπd2 (ca + cb )f8 (cb , ~r, t)f1 (ca , ~r, t)dcb 3 0 Z 2 ∞ + (ca + cb )f5 (cb , ~r, t)f4 (ca , ~r, t)dcb 3 0 Z 2 ∞ + (ca + cb )f3 (cb , ~r, t)f6 (ca , ~r, t)dcb 3 0 Z ∞ − (ca + cb )f7 (cb , ~r, t)f2 (ca , ~r, t)dcb 0

− −

1 3

Z

∞

1 3

Z

∞


0


0

1350007-11

(A.1)

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−

1 3

Z

∞

0


(A.2)


Direction 3 ∂f3 (ca , ~r, t) ∂f3 (ca , ~r, t) + ca~l3 ∂t ∂~r Z ∞ 2 = Cπd2 (ca + cb )f8 (cb , ~r, t)f1 (ca , ~r, t)dcb 3 0 Z 2 ∞ + (ca + cb )f5 (cb , ~r, t)f4 (ca , ~r, t)dcb 3 0 Z 2 ∞ (ca + cb )f2 (cb , ~r, t)f7 (ca , ~r, t)dcb + 3 0 Z ∞ − (ca + cb )f6 (cb , ~r, t)f3 (ca , ~r, t)dcb 0

1 3

Z

∞

1 3

Z

∞

1 − 3

Z

∞

− −


0


0


0

(A.3)


1 − 3

Z

∞

1 − 3

Z

∞

1 − 3

Z

∞


0


0


0

1350007-12

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Direction 5 ∂f5 (ca , ~r, t) ∂f5 (ca , ~r, t) + ca~l5 ∂t ∂~r Z ∞ 2 (ca + cb )f8 (cb , ~r, t)f1 (ca , ~r, t)dcb = Cπd2 3 0 Z 2 ∞ + (ca + cb )f3 (cb , ~r, t)f6 (ca , ~r, t)dcb 3 0 Z 2 ∞ + (ca + cb )f2 (cb , ~r, t)f7 (ca , ~r, t)dcb 3 0 Z ∞ − (ca + cb )f4 (cb , ~r, t)f5 (ca , ~r, t)dcb 0

1 3

Z

∞

1 − 3

Z

∞

1 − 3

Z

∞

−


0


0


0

(A.5)


1 − 3

Z

∞

1 − 3

Z

∞

1 − 3

Z

∞


0


0


0

1350007-13

(A.6)

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− − −

1 3

Z

∞

1 3

Z

∞

1 3

Z

∞


0


0

0


(A.7)


1 − 3

Z

∞

1 − 3

Z

∞

1 − 3

Z

∞


0


0


0

1350007-14

(A.8)

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Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 51006102) and the National Basic Research Program of China (No. 2011CB710705).


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