Accepted Manuscript

Space Time CE/SE and Discrete Ordinate Method for Solving Gas Flows of

Arbitrary Statistics

Bagus Putra Muljadi, Jaw-Yen Yang

PII: S0045-7930(12)00160-0

DOI: http://dx.doi.org/10.1016/j.compfluid.2012.04.021

Reference: CAF 1851

To appear in: Computers & Fluids

Received Date: 4 August 2011

Revised Date: 8 April 2012

Accepted Date: 17 April 2012

Please cite this article as: Muljadi, B.P., Yang, J-Y., Space Time CE/SE and Discrete Ordinate Method for Solving

Gas Flows of Arbitrary Statistics, Computers & Fluids (2012), doi: http://dx.doi.org/10.1016/j.compfluid.

2012.04.021

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Space Time CE/SE and Discrete Ordinate Method for Solving Gas Flows ofArbitrary Statistics

Bagus Putra Muljadia, Jaw-Yen Yanga,b,∗

aInstitute of Applied Mechanics, National Taiwan University, Taipei 10764, TAIWANbCenter of Quantum Science and Engineering, National Taiwan University, Taipei 10764, TAIWAN

1. Introduction

In recent years, due to the rapid advancements of microtechnology and nanotechnology, the characteristic lengthscales of the device or structure become comparable to the mean free path and the wavelength of energy and informa-tion carriers (mainly electrons, photons, phonons, and molecules), the classical continuum transport laws are no longerapplicable. It is generally believed that the microscopic description of Boltzmann equation (classical and semiclassi-cal) is adequate to treat transport phenomena in the mesoscale range. Besides, due to the different types of carriersmay involve simultaneously in a single problem, it is desirable to have a method that can allow one to treat them ina unified and parallel manner. With the semiclassical Boltzmann equation, it is possible to describe adequately themesoscale transport of particles of arbitrary statistics. The Bhatnagar-Gross-Krook relaxation time concept [1] can beapplicable to gases of particles of general statistics and is adopted in this study. The solution methodology developedin [2] can be applied to the present semiclassical Boltzmann-BGK equation in phase space. In this note, first, thediscrete ordinate method is used to discretize the velocity (or momentum or wave number) space in the semiclassicalBoltzmann-BGK equation into a set of equations in physical space with source terms that are independent of molec-ular velocity. Second, the resulting equations can be treated as scalar hyperbolic conservation laws with stiff sourceterms. Here, an explicit, second-order conservation element and solution element (CE/SE) scheme developed byChang [3, 4, 5] for solving conservation laws is adopted to describe the evolution of these equations. Due to the com-plexity of the transport problems of different particle statistics, an accurate yet simple numerical method exempt fromthe applications of Riemann solvers, flux splittings and monotonicity constraints, all exhibited by CE/SE is desired.The application of CE/SE on systems of distribution functions may exhibit different results when conventional upwindbased schemes are applied instead. This is due to its nontraditional features which includes: (i) a unified treatment ofspace and time; (ii) the introduction of conservation elements (CEs) and solution elements (SEs) as the vehicles forenforcing space-time flux conservation; (iii) exclusion of interpolation nor extrapolation procedure for computationof fluxes at an interface. Our goal is to first develop a 1D solver before considering a multi-dimensional approach.This is due to the demanding computational costs when 2D and 3D solvers are considered. We believe that thesefindings shall give a strong ground before advancing to multi-dimensional analysis eventually. In this note, numericalexperiments of one-dimensional semiclassical gas dynamical flows near low densities are presented to demonstratethe ability of the present algorithm.

2. Semiclassical Boltzmann-BGK Equation

The elements of semiclassical Boltzman-BGK equation are described as:(∂

∂t+

pm· ∇x − ∇U(x, t) · ∇p

)f (p, x, t) = −

f − f (0)

τ(1)

∗Corresponding author. Tel.: +886 2 3366 5636; fax: +8866 2 2362 9290Email address: yangjy@iam.ntu.edu.tw (Jaw-Yen Yang)

Preprint submitted to Computers and Fluids April 26, 2012

where m is the particle mass, U is the mean field potential and f (p, x, t) is the distribution function which representsthe average density of particles with momentum p at the space-time point x, t. Here, τ is the relaxation time and needsto be specified for each carrier scattering. The equilibrium distribution function for general statistics can be expressedas

f (0) (p, x, t) =1

z−1 exp[

p − m u(x, t)]2/2mkBT (x, t)

+ θ

(2)

where u(x, t) is the mean velocity, T (x, t) is temperature, kB is the Boltzmann constant and z (x, t) = exp (µ(x, t)/kBT (x, t))is the fugacity, where µ is the chemical potential. In (2), θ = −1, denotes the Fermi-Dirac statistics, θ = +1, the Bose-Einstein statistics and θ = 0 denotes the Maxwell-Boltzmann statistics. We note that even with the case of θ = 0, westill have chemical potential µ or fugacity z which is rather different from the usual classical Maxwellian distribution.The macroscopic dynamic variables of interest such as number density, momentum density and energy density are thefirst few low order moments of the distribution function and are defined by: n(x, t)

j(x, t)ε(x, t)

=∫

dph3

1p/m

p2/2m

f (p, x, t) (3)

where h is Planck’s constant. Other higher-order moments such as stress tensor Pi j and the heat flux vector Φi(x, t)can also be defined accordingly. The conservation laws of macroscopic properties can be obtained by multiplying Eq.(1) respectively by 1, p and p2/2m and integrating the resulting equations over all p. Consequently, the integrals ofthe collision terms in all three cases vanish automatically resulting in the conservation laws in differential equationsform for the conserved macroscopic quantities i.e., number density n(x, t), momentum density ρu(x, t), and energydensity ε(x, t) as follow:

∂n∂t

+∂

∂x j(nu j) = 0, (4)

∂ρui

∂t+

∂

∂x j(Pi j) = −ρ

∂

∂xiU, (5)

∂ε

∂t+

∂

∂x j(P jk · uk + Φ j) = −ρu · ∇xU, (6)

The transport coefficients such as viscosity η and thermal conductivity κ can be derived in terms of the relaxation timeas

η = τnkBT Q2(z)Q1(z) , (7)

κ = τ 5kB2m nkBT [ 7

2Q3(z)Q1(z) −

52Q2(z)Q1(z) ], (8)

where Qν(z) is the Fermi or Bose function of order ν given below. In the following, we will also consider the caseof semiclassical Euler limit in which the particle distribution function is always in equilibrium, i.e., f = f (0) andthe collision term of Eq. (1) vanishes automatically. In recent years, kinetic numerical methods for solving the idealquantum gas flows based on f = f (0) have been developed, see [6, 7]. Before we proceed, without losing generality,we neglect the influence of externally applied field U(x, t). To illustrate the present method, we formulate the generaldistribution in one spatial dimension as the following.

f (px, x, t) =1

z−1 exp[

px − mux]2/2mkBT (x, t)

+ θ

(9)

In a closed form in terms of quantum function, the macroscopic moments, i.e., number density n(x, t), momentumj(x, t), energy density ε(x, t), and pressure P(x, t) are given by

n(x, t) =

∫dpx

hf (0)(px, x, t) =

Q1/2(z)λ

(10)

2

j(x, t) =

∫dpx

hpx

mf (0)(px, x, t) = n(x, t)ux(x, t) (11)

ε(x, t) =

∫dpx

hpx

2

2mf (0)(px, x, t) =

Q3/2(z)2βλ

+12

mnu2x (12)

P(x, t) = nkBT (x, t)Q5/2(z)Q3/2(z)

(13)

where λ =√βh2

2πm is the thermal wavelength and β = 1/kBT (x, t). The functions Qυ(z) of order υ are respectivelydefined for Fermi-Dirac and Bose-Einstein statistics as

Fυ(z) ≡1Γ(υ)

∫ ∞

0dx

xυ−1

z−1ex + 1≈

∞∑l=1

(−1)l−1 zl

lυ(14)

Bυ(z) ≡1Γ(υ)

∫ ∞

0dx

xυ−1

z−1ex − 1≈

∞∑l=1

zl

lυ(15)

Here, Fυ(z) applies for Fermi-Dirac integral and Bυ(z) for Bose-Einstein’s, whereas Γ(υ) is gamma function. Thedefinition of macroscopic quantities in terms of Fermi or Bose function applies for both cases of quantum distributions.Before proceeding to the analysis of the integrals, we introduce the characteristic quantities of V∞ and t∞ for thepurpose of normalization. The characteristic velocity and time can be defined as,

V∞ =

√2kBT∞

mt∞ =

LV∞

(16)

with L defined as characteristic length of the problem. The normalized semiclassical Boltzmann-BGK equation,

∂ f (υx, x, t)∂t

+ υx∂ f (υx, x, t)∂x

= −f − f (0)

τ(17)

with particle velocity υx. From this part, our formulations are all considered normalized and the ‘hat’ sign is omittedfor simplicity.

3. Application of Discrete Ordinate Method

By applying the discrete ordinate method to Eq. (17), the quantum velocity distribution function in phase spacef (x, υx, t) can be rendered into a set of hyperbolic conservation equation with source terms in a physical space forfσ(x, t) where σ = 1, . . . , N. The resulting equations are,

∂ fσ(x, t)∂t

+ υσ∂ fσ(x, t)∂x

= −fσ − fσ(0)

τ(18)

with fσ and υσ represent the values of respectively f and υx evaluated at the discrete velocity points σ. Once we havesolved the discrete distribution function fσ(x, t), then the quadrature method is applied to calculate the macroscopicquantities. In this work, Newton-Cotes formula with adjustable intervals is utilized to cover wide range of velocity.We employ a variant of Newton-Cotes formula i.e., the composite five-point Boole’s rule, see [8],[9] and [10]. Therepeated Newton-Cotes formula over an interval [v1, vN] divided into N panels of width h = (vN − v1)/N is∫ vN

v1

f (v) dv =2

45h[7 f1 + 32 f2 + 12 f3 + 32 f4 + 14 f5

+ . . . + 32 fN−1 + 7 fN] −2(vN − v1)M(6)

945h6, (19)

where,M(i) = sups∈(v1,vN )

∣∣∣ f (i)(s)∣∣∣ . (20)

3

Here, f (i) denotes the ith order derivative.For every time level we can acquire and update the macroscopic moments in the physical space using a selected

quadrature method e.g.,

n(x, t) =∫ ∞

−∞

f (υx, x, t)dυx ≈ hN∑σ=1

Hσ fσ(x, t). (21)

4. Numerical Method

A space-time conservation element and solution element c − τc method with extended wiggle suppressing terms(known also as w − α scheme) is introduced to solve the scalar conservation law with stiff source terms. The acquire-ment of marching variables for 1 time level using CE/SE is divided into two half time steps. To be consisted with thenotation with the original work the time integration is carried out from (n − 1/2)th to nth time level, and subsequentlyfrom nth to (n + 1/2)th time level.The scheme used in this work is w − α scheme which can be written as

fσnj =

12

[(1 + ν) fσn−1/2j−1/2 + (1 − ν) fσ

n−1/2j+1/2 +

(1 − ν2)[( fσ x)n−1/2j−1/2 − ( fσ x)n−1/2

j+1/2 ]] +

∆t2

(−fσn−1

j − fσ(0)n−1j

τ). (22)

where( fσ x)n

j = (w−)nj ( fσ

τcx−)n

j + (w+)nj ( fσ

τcx+)n

j (23)

with the weighting functions (w±)nj defined as

(w±)nj = W±(( fσ

τcx−)n

j , ( fστcx+)n

j ;α). (24)

( fστcx−)n

j = (( fσ)nj − fσ

n−1/2j−1/2 − (2ν + 1 − τc) fσ x

n−1/2j−1/2 )/(1 + τc)

( fστcx+)n

j = ( fσn−1/2j+1/2 − (2ν − 1 + τc) fσ x

n−1/2j+1/2 − ( fσ)n

j )/(1 + τc). (25)

The parameter τc and α are chosen constants. In details, the step-by-step algorithm is explained as follow,

1. Initial information of fσ(n−1/2)j and ( fσx)(n−1/2)

j , ∀ j ∈ Ω are defined. Ω denote the set of all staggered space-time

mesh points ( j, n) in an Euclidian space E2. For every j ∈ Ω, the normalized term ( fσ x)(n−1/2)j can be calculated

from ( fσx)(n−1/2)j∈Ω using the relation ( fσ x)n

j = (∆x/4)( fσx)nj . At this initial state, the equilibrium values, fσ(0)n−1

j∈Ω

is defined the same as fσ(0)n−1/2j<Ω . Note that at (n − 1/2)th, j < Ω if ( j, n) ∈ Ω (see Fig. 2(a)).

2. After initialization in step 1, fσnj , ∀ j ∈ Ω, can be calculated using Eq. (22).

3. Eq. (23) is employed along with Eq. (24) to acquire the second marching variable ( fσ x)nj∈Ω, i.e., ( fσx)n

j∈Ω. ( fσ)nj

(calculated in step 2) and ( fσx)nj , ∀ j ∈ Ω are variables required in order to advance to (n + 1/2)th time level.

4. Step 2 shall be repeated to reach (n + 1/2)th time level from nth time level. For example, when calculatingfσ

n+1/2j+1/2 , Eq. (22) will be applied by shifting n→ (n + 1/2) and j→ ( j + 1/2) to obtain

fσn+1/2j+1/2 =

12

[(1 + ν) fσnj + (1 − ν) fσn

j+1 +

(1 − ν2)[( fσ x)nj − ( fσ x)n

j+1]] +

∆t2

(−fσ

n−1/2j+1/2 − fσ(0)n−1/2

j+1/2

τ). (26)

4

5. Step 3 is repeated; Eqs. (23) and (24) are employed (by shifting n → (n + 1/2) and j → ( j + 1/2)) to calculate( fσx)n+1/2

j+1/2 .

At this stage we have completed the integration of Eq. (18) for one full time step. The whole procedure shall berepeated to obtain the next time level. To be consistent with the molecular collision theory, the time step size (∆t) hasto be less than the local mean collision time τ. After acquiring the macroscopic moments from Eqs. (10), (11) and(12), we need to use the combination of these equations to update the values of z(x, t) and T (x, t). To solve z(x, t),which is the root of equation

Ψ1(z) = ε −Q3/2

4π(

nQ1/2

)3 −12

nu2x (27)

5. Computational Results

We chose to solve the one-dimensional gas flow problems near low densities. This problem was first proposedin [11] and is chosen for demonstrating the ability of the method to preserve the positivity of density and internalenergy. In this problem a diaphragm is located at x = 0.5, separating two regions of gases of arbitrary statistics;each remains in a constant equilibrium state at initial time t = 0. The macroscopic properties on both sides of thediaphragm are set different and given with (z,U,T ) = (0.446,−2, 1.6) for 0 ≤ x ≤ 0.5 and (z,U,T ) = (0.446, 2, 1.6)for 0.5 ≤ x ≤ 1. As the diaphragm is released, strong rarefaction waves propagate toward opposite directions. InFig. 1, the plots of density, pressure, energy, fugacity, temperature and velocity are given for the three statistics. Thegiven initial condition correspond to different values of densities, pressures and energies for the three statistics ascan be seen in the figure. In this example, 300 grid points are used with 800 Newton-Cotes discrete velocity points.τ = 0.001 is used with Courant number of 0.1. CE/SE scheme with α = 3 is utilized. In Fig. 2, the range of constantrelaxation times are tested to test the robustness of the algorithm. For the case of Maxwell-Boltzmann gas, we testedthe values of τ = 0.001, 0.01, 0.1 as well as the case of f = f (0) by which the solution shall be regarded as that ofEuler solution. In this example 400 grid points are used with 900 Newton-Cotes discrete velocity points. For the caseof Maxwell-Boltzmann statistic, the given initial condition correspond to n = 1 and P = 0.4 for 0 ≤ x ≤ 1. The resultsfor Maxwell-Boltzmann gas in Euler limit can be compared with the existing works, see [11, 12]. Note that with thecurrent method it is possible to plot fugacity for Maxwell-Boltzmann gas.

6. Concluding Remarks

A direct algorithm constructed to solve semiclassical Boltzmann-BGK equation for particles of all statistics isapplied on a one-dimensional gas flow problems utilizing the application of CE/SE scheme. When Fermi-Diracor Bose-Einstein statistics are considered, we do not have experimental nor computational results yet to compareour results to, especially at transitional regions. However, there are adequate results to compare to when Maxwell-Boltzmann statistic is considered, especially at Euler limit. Therefore quantitative comparisons can possibly be made.Taking into account that all the statistics are treated on an equal foot and on parallel manner, we could be at leastassured that the current method and the generated results has sound theoretical ground.Note that CE/SE scheme is formed by two coupled discrete equations (i.e., Eqs (22) and (23)) involving independentnumerical variables f and ∂ f /∂x; Therefore, when multidimensional case and complex boundary condition e.g.,including wall reflection are considered, special treatment at the boundary has to be done in order to maintain a soundphysics. This treatment might be unique in regard to the application of CE/SE scheme.Outside the scope of this note, the extension to multi-dimensional cases covering more complex geometries withboundary value problems using CE/SE will be subject to a further study.

7. Acknowledgements

This work is done under the auspices of National Science Council, TAIWAN through grants NSC-99-2922-I-606-002.

5

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Figure 1: Macroscopic properties of Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein gases

7

Figure 2: Macroscopic properties of Maxwell-Boltzmann gases at τ = 0.1, 0.01, 0.001 and the Euler solution

8