A higher-order moment method of the lattice Boltzmann model for the conservation law equation

Applied Mathematical Modelling 34 (2010) 481–494

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

A higher-order moment method of the lattice Boltzmann modelfor the conservation law equation q

Yinfeng Dong, Jianying Zhang, Guangwu Yan *

College of Mathematics, Jilin University, Changchun 130012, PR China

a r t i c l e i n f o

Article history:Received 21 October 2007Received in revised form 6 May 2009Accepted 1 June 2009Available online 16 June 2009

Keywords:Lattice Boltzmann modelHigher-order moment methodConservation law equation

0307-904X/$ - see front matter � 2009 Published bdoi:10.1016/j.apm.2009.06.024

q This work is Project 20092005 supported by G90305013), and the Chuangxin Foundation of Jilin U

* Corresponding author.E-mail address: [email protected] (G. Yan

a b s t r a c t

In this paper, we proposed a higher-order moment method in the lattice Boltzmann modelfor the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK)model, the higher-order moment method has a wide flexibility to select equilibrium distri-bution function. This method is based on so-called a series of partial differential equationsobtained by using multi-scale technique and Chapman–Enskog expansion. According toHirt’s heuristic stability theory, the stability of the scheme can be controlled by modulatingsome special moments to design the third-order dispersion term and the fourth-order dis-sipation term. As results, the conservation law equation is recovered with higher-ordertruncation error. The numerical examples show the higher-order moment method canbe used to raise the accuracy of the truncation error of the lattice Boltzmann scheme forthe conservation law equation.

� 2009 Published by Elsevier Inc.

1. Introduction

In recent years, computational methods based on the lattice Boltzmann method (LBM) have attracted much attention.They have been developed as an alternative method for computational fluid dynamics (CFD). These lattice Boltzmann meth-ods originated from a Boolean fluid model known as the lattice gas automata (LGA) [1] originally developed to overcomecertain drawbacks such as the presence of statistical noise and lack of Galilean invariance of LGA for modeling fluid flowbased upon kinetic theory [2–5]. The lattice Boltzmann models have been used to simulate formidable problems such asmultiphase flows [6–11], multi-component flows [7,12–14], porous media flows [15,16], flows of suspensions [17,18], andcompressible flows [19–28]. Additionally, the lattice Boltzmann models have been developed to simulate linear and nonlin-ear partial differential equations such as wave motion equation [29], Burgers equation [30], KdV equation [31], Lorenz equa-tions [32], the shallow equation [33], the Richards equation [34], Poisson equation [35], and the nonlinear Schrödingerequation [36–40].

Unlike conventional methods based on macroscopic continuum equation, these LBMs start from mesoscopic kinetic equa-tion, say, the lattice Boltzmann equation, to determine macroscopic fluid flows. Their kinetic nature brings certain advanta-ges over conventional numerical methods, such as their algorithmic simplicity, parallel computation, easy handing ofcomplex boundary conditions, and efficient numerical simulations. Especially their algorithmic simplicity and the flexibilityto select equilibrium distribution functions are outstanding advantages.

There are two basic problems in the lattice Boltzmann method: (1) how to improve the accuracy of lattice Boltzmannmodel, (2) how to obtain higher-order accuracy boundary conditions for some complex flows. In this paper, we focus on

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radate Innovation Fund of Jilin University, National Nature Science Foundation of China (Grant No.niversity (No. 2004CX041).

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482 Y. Dong et al. / Applied Mathematical Modelling 34 (2010) 481–494

the first problem. Furthermore, we choose one of the conversation law equations the mass conservation equation to con-struct a high accuracy LBM model. The one-dimensional mass conservation equation is

@u@tþ @FðuÞ

@x¼ 0; ð1Þ

where u expresses mass, F(u) is the mass flux. In Eq. (1), the mass flux F(u) is a nonlinear function of mass u. The variable ucan also express momentum and energy, and then flux F(u) represents impulse flux and energy flux respectively.

In order to propose a higher-order accuracy model for the conservation law equation, the key step is to supply a series ofpartial differential equations in different time scales based on the Chapman–Enskog analysis [41–44].

The Chapman–Enskog analysis consists of the Chapman–Enskog expansion and multi-scale time and multi-scale spacetechniques. We can obtain the macroscopic equations by combining the equations in the different scales [41,42]. In Ref.[43], we introduce four partial differential equations to construct a LBM model. However, the four partial differential equa-tions are not competent for constructing the LBM model with the higher-order dispersion term and dissipation term. A de-tailed analysis for the generalized lattice Boltzmann equation with multiple-relaxation-time collision operators is proposedby Ginzburg [44]. In Ref. [44], the model works with the multiple-relaxation-time collision operators (MRT), two-relaxation-time collision operators (TRT) and the Bhatnagar–Gross–Krook collision operator (BGK), some types of partial differentialequations are discussed. In Ref. [44], the fourth and sixth order analysis of the diffusion term and third-order analysis ofthe convection term are given. In the frame of the TRT models in 2D and 3D, Ginzburg presents the third order analysisof the convection term and the fourth and sixth order analysis of the diffusion term. The approach is the only possibleone, at least beyond the BGK model. The leading-order correction vanishes when,

43

12þ 1

ke

� �12þ 1

kD

� �¼ 1

9;

where ke and kD are two eigenvalues of the TRT model. In the case of the BGK, 1ke¼ 1

kD¼ �s the solution gives s� ¼ 1

2þ 12�1=2.This is similar to C3 = 0 in our model.

In this paper, we furnished more higher-order multi-scale time expansion than the preceding work in Ref. [29]. It is dif-ficult to obtain the lattice Boltzmann equation in any time scale. Therefore, we only provide a series of six partial differentialequations in this paper. In our former works the equilibrium function for the Navier–Stokes equations up to fourth ordermoments in the frame of the two and three-dimensional problems [45,46], in the paper, we proposed the moments up tosixth order.

Two open issues on the lattice Boltzmann method are considered in this paper. The first issue concerns the approach ofhow to obtain a higher-order truncation error for a lattice Boltzmann model. If the multi-scale technique and the Chapman–Enskog expansion are used, then how many series of partial differential equations are enough, and how many conservationlaws in different time scales are enough? The second issue concerns numerical convergence, numerical stability of latticeBoltzmann scheme and its dependence on the Knudsen number e and relaxation time factor s and how to construct thethird-order dispersion term and the fourth-order dissipation term.

This paper is organized as follows: in the next section, a series of partial differential equations are given. In Section 3, wepropose a lattice Boltzmann model for one-dimensional conservation law equation by using higher-order moment method.In Section 4, some numerical examples are given, and in Section 5 some conclusions are discussed.

2. A series of partial differential equations in different time scales in the lattice Boltzmann model

2.1. The lattice Boltzmann equation

Consider a D-dimensional lattice meshes, the particles velocity can be discrete into b directions. In the lattice, each nodehas b nearest neighbors connected by b links. At each node, it allows rest particles to exist, thus, the particles velocity is dis-crete into b + 1 velocities actually. The distribution function fa (x, t) is defined as the one-particle distribution function withvelocity ea at time t, position x.

We assume fa(x, t) possesses the equilibrium distribution f eqa ðx; tÞ, and it meets the condition
X
af eqa ðx; tÞ ¼

Xa

faðx; tÞ: ð2Þ

The distribution function fa(x, t) satisfies the lattice Boltzmann equation

faðxþ ea; t þ 1Þ ¼ faðx; tÞ þXaðx; tÞ; ð3Þ

where s is the single-relaxation time factor, Xa(x,t) is the collision term. In Eq. (3), collision term Xa(x, t) can be expressed asmultiple-relaxation-time collision operators, two-relaxation-time collision operators and the Bhatnagar–Gross–Krook colli-sion operator [44]. In those models, the BGK collision operator is the simplest one [2]. The BGK collision operator is

Xaðx; tÞ ¼ �1s

faðx; tÞ � f eqa ðx; tÞ

� �; ð4Þ

Y. Dong et al. / Applied Mathematical Modelling 34 (2010) 481–494 483

where f eqa ðx; tÞ is the local equilibrium distribution function at position x, and time t, with velocity ea, s is the single-relax-

ation-time.

2.2. A series of partial differential equations in different time scales

The Knudsen number e is defined as e ¼ ‘L, where ‘is the mean free path, and L is the characteristic length. We assume that

the Knudsen number e is equal to the time step Dt[29], thus, the lattice Boltzmann equation (3) can be written as

faðxþ eea; t þ eÞ ¼ faðx; tÞ �1s

faðx; tÞ � f eqa ðx; tÞ

� �: ð5Þ

Using the Taylor expansion to Eq. (5),

faðxþ eea; t þ eÞ � faðx; tÞ ¼X1n¼1

en

n!

@

@tþ ea

@

@x

� �n

faðx; tÞ: ð6Þ

Retaining terms up to O(e7), we have

fa xþ eea; t þ eð Þ � fa x; tð Þ ¼ e@

@tþ ea

@

@x

� �faðx; tÞ þ

e2

2 @

@tþ ea

@

@x

� �2

faðx; tÞ þe6

3 @

@tþ ea

@

@x

� �3

faðx; tÞ

þ e4

24@

@tþ ea

@

@x

� �4

faðx; tÞ þe5

120@

@tþ ea

@

@x

� �5

faðx; tÞ þe6

720@

@tþ ea

@

@x

� �6

faðx; tÞ þ Oðe7Þ:

ð7Þ

Next step is that Chapman–Enskog expansion [47] is applied to fa(x, t) under the assumption of small Knudsen number e, it is

fa ¼ f eqa þ

X6

i¼1

eif ia þ Oðe7Þ: ð8Þ

To discuss changes in different time scales, introduced as t0, . . . , t6,

ti ¼ eit; i ¼ 0; . . . ;6 ð9Þ

and

@

@t¼X6

i¼0

ei @

@tiþ Oðe7Þ: ð10Þ

Substituting Eqs. (8)–(10) into Eq. (7),the equation to order e is

Df ð0Þa ¼ �1s

f ð1Þa ; ð11Þ

where f ð0Þa � f eqa , partial differential operator D � @

@t0þ ea

@@x.

The equation to order e2 is

@

@t1f ð0Þa þ

12� s

� �D2 f ð0Þa ¼ �1

sf ð2Þa : ð12Þ


s2 � sþ 16

� �D3f ð0Þa þ 2

12� s

� �D@

@t1f ð0Þa þ

@

@t2f ð0Þa ¼ �1

sf ð3Þa : ð13Þ


�s3 þ 32s2 � 7

12sþ 1

24

� �D4f ð0Þa þ 3 s2 � sþ 1

6

� �D2 @

@t1f ð0Þa þ 2

12� s

� �D@

@t2f ð0Þa þ

@

@t3f ð0Þa þ

12� s

� �@2

@t21

f ð0Þa ¼ �1s

f ð4Þa :

ð14Þ


s4 � 2s3 þ 54s2 � 1

4sþ 1

120

� �D5f ð0Þa þ 4 �s3 þ 3

2s2 � 7

12sþ 1

24

� �D3 @

@t1f ð0Þa þ 3 s2 � sþ 1

6

� �D2 @

@t2f ð0Þa

þ 212� s

� �D@

@t3f ð0Þa þ

@

@t4f ð0Þa þ 3 s2 � sþ 1

6

� �D@2

@t21

f ð0Þa þ 212� s

� �@2

@t1@t2f ð0Þa ¼ �1

sf ð5Þa : ð15Þ



�s5 þ 52s4 � 13

6s3 þ 3

4s2 � 31

360sþ 1

720

� �D6f ð0Þa þ 5 s4 � 2s3 þ 5

4s2 � 1

4sþ 1

120

� �D4 @

@t1f ð0Þa

þ 4 �s3 þ 32s2 � 7

12sþ 1

24

� �D3 @

@t2f ð0Þa þ 3 s2 � sþ 1

6

� �D2 @

@t3f ð0Þa þ 2

12� s

� �D@

@t4f ð0Þa þ

@

@t5f ð0Þa

þ 6 �s3 þ 32s2 � 7

12sþ 1

24

� �D2 @

2

@t21

f ð0Þa þ 6 s2 � sþ 16

� �D

@2

@t1@t2f ð0Þa þ 2

12� s

� �@2

@t1@t3f ð0Þa

þ s2 � sþ 16

� �@3

@t31

f ð0Þa þ12� s

� �@2

@t22

f ð0Þa ¼ �1s

f ð6Þa : ð16Þ

Eqs. (11)–(16) is so-called a series of partial differential equations in different time scales. It is suitable for one-dimensional,two-dimensional and three-dimensional cases. In Ref. [29], a series of four partial differential equations (11)–(14) are given.We find that Eqs. (11)–(14) are not enough to find higher-order moment. By increasing the number of the series of partialdifferential equations, we can obtain the model with higher-order truncation error, the third- order dispersion term and thefourth-order dissipation term. Even the fifth-order dispersion term and the sixth-order dissipation term can be obtained.

We also find six polynomials of the relaxation time factor s in Eqs. (11)–(16), they are

C1 ¼ 1; ð17Þ

C2 ¼12� s; ð18Þ

C3 ¼ s2 � sþ 16¼ C2

2 �1

12; ð19Þ

C4 ¼ �s3 þ 32s2 � 7

12sþ 1

24¼ C3

2 �16

C2; ð20Þ

C5 ¼ s4 � 2s3 þ 54s2 � 1

4sþ 1

120¼ C4

2 �14

C22 þ

1120

; ð21Þ

C6 ¼ �s5 þ 52s4 � 13

6s3 þ 3

4s2 � 31

360sþ 1

720¼ C5

2 �13

C32 þ

17720

C2: ð22Þ

Polynomials ((17)–(22)) are the first six Bernoulli polynomials, which are in full agreement with the results in the literature[48]. They can be used to indicate coefficients of the dispersion term and the dissipation term to the modified conservationlaw equation. These polynomials (Eqs. (17)–(22)) have the character: when s > 1.0,C2, C4 and C6 are negative numbers, but C3

and C5 are positive numbers. In Fig. 1, we point out the relations between Ci and the relaxation time factor s.

2.3. Definitions for moments of equilibrium distribution functions

The macroscopic quantity u(x, t) is defined by

uðx; tÞ ¼X

afaðx; tÞ: ð23Þ

Fig. 1. Relations between coefficients Ci ands.


According to Eq. (2), we have
Xa
f eqa ðx; tÞ ¼ uðx; tÞ: ð24Þ

Bases on Chapman–Enskog expansion, thus
Xa
f ðnÞa ðx; tÞ ¼ 0; n P 1: ð25Þ

Combining these equations and higher-order moments, the conservation laws in different time scales are given in Appendix A.Some moments of the equilibrium distribution function are defined as following:

uðx; tÞ ¼X

af ð0Þa ðx; tÞ; ð26Þ

mð0Þj ðx; tÞ ¼X

af ð0Þa ðx; tÞeaj; ð27Þ

pð0Þij ðx; tÞ ¼X

af ð0Þa ðx; tÞeaieaj; ð28Þ

Pð0Þijk ðx; tÞ ¼X

af ð0Þa ðx; tÞeaieajeak; ð29Þ

Q ð0Þijkmðx; tÞ ¼X

af ð0Þa ðx; tÞeaieajeakeam; ð30Þ

Rð0Þijkmnðx; tÞ ¼X

af ð0Þa ðx; tÞeaieajeakeamean; ð31Þ

Sð0Þijkmnlðx; tÞ ¼X

af ð0Þa ðx; tÞeaieajeakeameaneal: ð32Þ

Especially, for one-dimensional model, these moments are defined as

uðx; tÞ ¼X

afaðx; tÞ; ð33Þ

m0ðx; tÞ ¼X

afaðx; tÞea; ð34Þ

p0ðx; tÞ ¼X

afaðx; tÞe2

a; ð35Þ

P0ðx; tÞ ¼X

afaðx; tÞe3

a; ð36Þ

Q 0ðx; tÞ ¼X

afaðx; tÞe4

a; ð37Þ

R0ðx; tÞ ¼X

afaðx; tÞe5

a; ð38Þ

S0ðx; tÞ ¼X

afaðx; tÞe6

a: ð39Þ

For some velocity sets in common use, these equilibrium distribution functions can be solved. In the Appendix B, the equi-librium distribution functions for the 3-bit model, 5-bit model, and 7-bit model are given respectively. According to the def-inition of these moments, we obtain some conservation laws in different time scales, see Appendix C.

3. The conservation law equation with fourth-order truncation error

3.1. Conservation law equation

Let us consider the one-dimensional conservation law equation (1), the flux F(u) is a nonlinear function of u. It means thatmass diffusion vanishes [49–52], that is to say, the diffusion coefficient l equals to zero.

@u@tþ @FðuÞ

@x¼ l @

2u@x2 ; l ¼ 0: ð40Þ

In order to recover the conservation law equation, the flux F(u) meets

FðuÞ ¼ m0 ¼X

af ð0Þa ea: ð41Þ

Therefore, the conservation law in the time scale t0 is

@u@t0þ @FðuÞ

@x¼ 0: ð42Þ


Eq. (42) can be written as

@u@t0¼ � @FðuÞ

@x¼ � @FðuÞ

@u@u@x: ð43Þ

3.2. Looking for p0 and P0

Summing the Eqs. (11)–(14) over a and making (11) + (12)*e + (13)*e2 + (14)*e3, we obtain

@u@tþ @FðuÞ

@xþ eC2

Xa

D2f ð0Þa þX

ae2C3D

3f ð0Þa þ e3C2@2u@t2

1

þX

a3e3C3D

2 @f ð0Þa

@t1þXa

e3C4D4f ð0Þa ¼ Oðe4Þ: ð44Þ

From Eq. (43), we have

@FðuÞ@u

@u@t0þ @FðuÞ

@u

� �2@u@x¼ 0: ð45Þ

If p0 and P0 meet the following conditions:

@p0

@u¼ @FðuÞ

@u

� �2

ð46Þ

and

@P0

@u¼ @FðuÞ

@u

� �3

; ð47Þ

then

Xa

D2f ð0Þa ¼ @

@t0

@u@t0þ @FðuÞ

@x

� �þ @

@x@FðuÞ@t0

þ @p0

@x

� �¼ 0; ð48Þ

Xa

D3f ð0Þa ¼ @2

@t20

@u@t0þ @FðuÞ

@x

� �þ 2

@2

@x@t0

@FðuÞ@t0

þ @p0

@x

� �þ @2

@x2

@p0

@t0þ @P0

@x

!¼ 0: ð49Þ

The Eq. (44) becomes

@u@tþ @FðuÞ

@xþ e3C2

@2u@t2

1

þX

a3e3C3D

2 @f ð0Þa

@t1þX

ae3C4D

4f ð0Þa ¼ Oðe4Þ: ð50Þ

3.3. Finding the e3 terms

The e3 terms are e3C2@2u@t2

1,P

a3e3C3D2 @f ð0Þa@t1

, andP

ae3C4D4f ð0Þa . According to Eqs. (12) and (48), we have

@u@t1¼ �

Xa

C2D2f ð0Þa ¼ 0;

@2u@t2

1

¼ 0: ð51Þ

Other term is

Xa

e3C4D4f ð0Þa ¼ e3C4

@3

@t30

Xa

Df ð0Þa þ 3@3

@t20@x

@F@t0þ @p

0

@x

� �þ 3

@3

@t0@x2

@p0

@t0þ @P0

@x

!þ @3

@x3

@P0

@t0þ @Q 0

@x

!" #

¼ e3C4@3

@x3

@P0

@t0þ @Q0

@x

!: ð52Þ

If we select that Q0 meets

@Q0

@u¼ @FðuÞ

@u

� �4

; ð53Þ

then

Xa

e3C4D4f ð0Þa ¼ C4e3 @3

@x3

@P0

@t0þ @Q0

@x

!¼ 0: ð54Þ


The Eq. (50) becomes

@u@tþ @FðuÞ

@x¼ Oðe4Þ: ð55Þ

Summing the Eqs. (11)–(16) over a and making (11) + (12)*e + (13)*e2 + (14)*e3 + (15)*e4 + (16)*e5, the modified conserva-tion law equation is

@u@tþ @FðuÞ

@x¼ E4 þ E5 þ Oðe6Þ; ð56Þ

where E4, E5 are

E4 ¼ �e4X

aC5D

5f ð0Þa þ 4C4D3 @

@t1f ð0Þa þ 3C3D

2 @

@t2f ð0Þa þ 2C2D

@

@t3f ð0Þa þ 3C3D

@2

@t21

f ð0Þa þ 2C2@2

@t1@t2f ð0Þa

" #; ð57Þ

E5 ¼ �e5X

a

"C6D

6f ð0Þa þ 5C5D4 @

@t1f ð0Þa þ 4C4D

3 @

@t2f ð0Þa þ 3C3D

2 @

@t3f ð0Þa þ 2C2D

@

@t4f ð0Þa

þ6C4D2 @

2

@t21

f ð0Þa þ 6C3D@2

@t1@t2f ð0Þa þ 2C2

@2

@t1@t3f ð0Þa þ C3

@3

@t31

f ð0Þa þ C2@2

@t22

f ð0Þa

#: ð58Þ

3.4. The dispersion term

According toP

aDf ð0Þa ¼ 0,P

aD2f ð0Þa ¼ 0,

PaD

3f ð0Þa ¼ 0, andP

aD4f ð0Þa ¼ 0, the dispersion term is

E4 ¼ �X

ae4C5D

5f ð0Þa ¼ �e4C5@4

@x4

@Q0

@t0þ @R0

@x

!: ð59Þ

If assume that

@R0

@u¼ @FðuÞ

@u

� �5

þ k; ð60Þ

then the fifth-order dispersion term is

E4 ¼ �X

ae4C5D

5f ð0Þa ¼ �e4C5@4

@x4

@Q0

@t0þ @R0

@x

!¼ �e4kC5

@5u@x5 ; ð61Þ

where k is a parameter.

3.5. The dissipation term

In the dissipation term, some terms are zero except for the term �P

ae5C6D6f ð0Þa . It is

�Xa

e5C6D6f ð0Þa ¼ �e5C6 5

@5

@t0@x4 k@u@x

� �þ @5

@x5

@R0

@t0þ @S0

@x

!" #: ð62Þ

Therefore

�Xa

e5C6D6f ð0Þa ¼ �e5C6

@5

@x5 �5k@F@xþ @R0

@t0þ @S0

@x

!: ð63Þ

If we select that

�5k@F@xþ @R0

@t0þ @S0

@x¼ l @u

@x; ð64Þ

then S0 meets

@S0

@u¼ lþ 6k

@F@uþ @F

@u

� �6

: ð65Þ

Lastly, we obtain the conservation law equation with the fifth-order dispersion term and the sixth-order dissipation term

@u@tþ @FðuÞ

@x¼ �e4kC5

@5u@x5 � e5lC6

@6u@x6 þ Oðe6Þ: ð66Þ


In summary, higher-order moments p0, P0, Q0, R0 and S0 meet Eqs. (46), (47), (53), (60) and (65) respectively. When F(u) is aspecific function, the higher-order moments above should be solved.

3.6. The Hirt’s heuristic stability conditions

In Eq. (66), the fifth dispersion is �e4kC5@5u@x5 , the sixth dissipation term is �e5lC6

@6u@x6 . According to Hirt’s heuristic stability

theory [53,54], coefficients e4kC5 > 0, e5lC6 < 0 are necessary stability conditions. Therefore, the stability of the lattice Boltz-mann scheme Eq. (3) is controlled by the condition e4kC5 > 0 and e5lC6 < 0. Select k > 0, l > 0, and s > 1, then e4kC5 > 0 ande5lC6 < 0.

4. Numerical examples

Since our purpose is to design the lattice Boltzmann model for higher- order accuracy of truncation error by using higher-order method, we will present the numerical examples in this section which are on linear equations and on nonlinear con-servation law equations. All the examples are standard problems widely used in the literature to test various schemes forthese sample equations.

In these numerical examples, we use ‘‘macroscopic variable” boundary conditions. Three kinds of boundary conditionsare:

(1) Dirichlet boundary condition, at two ends, u(0, t) = uL, u(M, t) = uL, where M is the lattice size. Then,

f tþ1a ð0; tÞ ¼ f eq

a ð0; tÞ; f tþ1a ðM; tÞ ¼ f eq

a ðM; tÞ;

where f eqa ð0; tÞ and f eq

a ðM; tÞ are equilibrium distribution functions at time t on x = 0 and x = M, calculated by using therelations of equilibrium distribution function and macroscopic variable u.

(2) Neumann boundary condition, at two ends, @@x uð0; tÞ ¼ 0, @

@x uðM; tÞ ¼ 0. We use u(0, t) = u(Dx, t), u(M, t) = u(M � Dx, t),where Dx is the spatial step. Then,


a ðDx; tÞ; f tþ1a ðM; tÞ ¼ f eq

a ðM � Dx; tÞ;

where f eqa ðDx; tÞ and f eq

a ðM � Dx; tÞ are equilibrium distribution functions at time t on x = Dx and x = M � Dx, calculatedby using the relations of equilibrium distribution function and macroscopic variable u.

(3) Periodic boundary condition: In this paper, periodic boundary condition is

uð0; tÞ ¼ uðM � 2; tÞ;uð1; tÞ ¼ uðM � 1; tÞ; uð2; tÞ ¼ uðM; tÞ;

where x = 0,1, . . . ,M, then,


a ðM � 2; tÞ; f tþ1a ðM; tÞ ¼ f eq

a ð2; tÞ;

where f eqa ðM � 2; tÞ and f eq

a ð2; tÞ are equilibrium distribution functions at time t on x = M � 2 and x = 2, calculated byusing the relations of equilibrium distribution function and macroscopic variable u.

Example 1

@u@t þ @u

@x ¼ 0; 0 6 x 6 1 t > 0

uðx;0Þ ¼ 1; 0:35 6 x 6 0:75 otherwise uðx; 0Þ ¼ 0;

uð0; tÞ ¼ uð1; tÞ ¼ 0; t > 0;

8><>: ð67Þ

Example 2

@u@t þ @u

@x ¼ 0; 0 6 x 6 1 t > 0

uðx;0Þ ¼ ð1� ½ði� 50Þ=15�2Þ12 35 6 i 6 65;where x ¼ i=100

0 otherwise

(

uð0; tÞ ¼ uðm; tÞ; t > 0:

8>>>>><>>>>>:

ð68Þ

Examples 1 and 2 are one-dimensional linear conservation equations [55]. In Figs. 2a and 3a, we give the comparisons ofexact solution and the LBM results. We also present the absolute error curves in Figs. 2b and 3b. These absolute errors arecontrolled in the scope of (�5.0 � 10�5,5.0 � 10�5). Especially, we find that this model has higher resolution in the discon-tinuous regions. Since this model possesses fourth-order accuracy of truncation error, the discontinuity should spread overone grid mesh only.

Fig. 2. Comparison of exact solution and the LBM result of the Example 1. (a) Comparison of exact solution and the LBM result. (b) Error bars of the latticeBoltzmann model. Parameters are: c = 500.0, lattice size M = 100, time t = 10,000Dt = 0.2, k = 0.1, l = 0.001.

Fig. 3. Comparison of exact solution and the LBM result of the Example 2 at time t = 0.4. (a) Comparison of exact solution and the LBM result. (b) Error of thelattice Boltzmann model. Parameters are: c = 500.0, lattice size M = 100, time t = 20,000, Dt = 0.4, k = 0.1, l = 0.001.


Example 3

@u@t þ u @u

@x ¼ 0; �0:5 6 x 6 0:5 t > 0

uðx;0Þ ¼ 1 x < 0:50 x P 0:5

�uð0; tÞ ¼ uð1; tÞ ¼ 0; t > 0:

8>><>>: ð69Þ

We also apply our model to the one-dimensional nonlinear conservation law equations [52]. In Example 3, we obtain anumerical result with higher resolution (see Fig. 4a). The absolute error curve is plotted in Fig. 4b. We find the error is lessthan 0.002.

The last example is Eq. (70), the smooth 1-periodic initial data is

uðx;0Þ ¼ SinðpxÞ: ð70Þ

The periodic boundary condition u(0,t) = u(1,t) is used. In order to obtain the exact numerical result, we use the fourth-order Runge–Kutta method to solve the Eq. (70). It is well-known that solution of Eq. (70) develops a shock discontinuity att � 0.31. In Fig. 5 we give two numerical results in time t = 0.15 and t = 0.4. At time t = 0.15 the shock begins to shapes, and at

Fig. 4. Comparison of exact solution and the LBM result of the Example 1 at time t = 0.2. (a) Comparison of exact solution and the LBM result. (b) Absoluteerror of the lattice Boltzmann model. Parameters are: c = 500.0, lattice size M = 100, time t = 10,000, Dt = 0.2, k = 0.1, l = 0.001.

Table 1L1-norm of the errors for Numerical Solutions of the Example 4. In LBM solutions, time steps N = 40, N = 80, N = 160, N = 320 are corresponding to M = 54,M = 108, M = 216 and M = 432 for t = 0.15. Time steps N = 40, N = 80, N = 160, N = 320 are corresponding to M = 20, M = 40, M = 80 and M = 160 for t = 0.4.

t = 0.15 t = 0.4

N LxF ORD STG LBM LxF ORD STG LBM

The Eq. (44) becomes 40 0.023702 0.02620 0.000859 0.000437 0.044449 0.003612 0.000849 0.00042280 0.12249 0.000667 0.000232 0.0000748 0.023486 0.001291 0.000277 0.0000851160 0.06246 0.000169 0.000061 0.000034 0.011393 0.000498 0.000098 0.000045320 0.003158 0.000043 0.000016 0.000005 0.005235 0.000209 0.000038 0.000003

Fig. 5. Comparison between exact solution and the LBM result of the Example 4. Solid line is the exact solution, circles are this LBM result. (a) t = 75,Dt = 0.15, (b) t = 200, Dt = 0.4. Parameters are: k = 0.1, l = 0.001, c = 5.0, lattice size M = 100.


time t = 0.4 the shock has formed. Table 1 shows the L1-norm of the errors at the pre-shock time t = 0.15 and post-shock timet = 0.4 for some classical numerical methods and this lattice Boltzmann model [52], in which the L1-norm is defined as

Table 2Convergence rates for Numerical Solutions of the Example 4. In LBM solutions, time steps N1 = 40, N2 = 80, N3 = 160, N4 = 320 are corresponding to M1 = 54,M2 = 108, M3 = 216 and M4 = 432 for t = 0.15. Time steps N1 = 40, N2 = 80, N3 = 160, N4 = 320 are corresponding to M1 = 20, M2 = 40, M3 = 80 and M4 = 160 fort = 0.4.

t = 0.15 t = 0.4

LxF ORD STG LBM LxF ORD STG LBM

R1 �1.7 � 10�5 1.2 � 10�5 3.5 � 10�7 1.8 � 10�7 4.1 � 10�5 3.7 � 10�6 8.7 � 10�7 4.8 � 10�7

R2 1.1 � 10�5 6.6 � 10�8 2.1 � 10�8 6.2 � 10�9 5.6 � 10�6 3.2 � 10�7 7.1 � 10�8 1.9 � 10�8

R3 1.8 � 10�6 4.3 � 10�9 1.6 � 10�9 1.2 � 10�9 6.8 � 10�7 3.1 � 10�8 6.1 � 10�9 3.4 � 10�9


L1 ¼Pm

i¼1jErij, M is the lattice size. In Table 1, we find this lattice Boltzmann model has higher accuracy and resolutionthan the first-order Lax–Friedrich scheme (LxF), the second-order non-oscillatory central differencing non-staggeredscheme (ORD), and the second-order non-oscillatory central differencing with staggered scheme (STG). Our methodhas two advantages, which are: (1) the widths of the shock waves are only one grid that are less than the corresponding clas-sical high-resolution model. (2) The absolute errors are in the scope of (�10�4,10�4). These are the features we are seekingfor.

In order to show the convergence effect, we define the convergence rate

Ri ¼LðiÞ1Mi� Lðiþ1Þ

1Miþ1

Niþ1 � Ni; ð71Þ

where LðiÞ1 ,Mi, and Ni denote L1-norm, the lattice size, the time steps for case i, respectively. In the Table 2, we find the LBMscheme possesses higher convergence rate than the LxF, ORD and STG schemes.

Example 4

@u@t þ u @u

@x ¼ 0; 0 6 x 6 1 t > 0

uðx;0Þ ¼ SinðpxÞ 0 6 x 6 1

uð0; tÞ ¼ uð1; tÞ t > 0

8>><>>: ð72Þ

5. Conclusion

In 1991, Frisch pointed out the possibility that increases the requirements of higher-order moment may be used toconstruct lattice gas model for Navier–Stokes equation [56]. According to this idea, we have completed the higher-ordermoment method for the conservation law equation. The numerical results agree well with the results of classicalschemes.

Some key points are as follows:

(1) We obtain a new series of partial differential equations in the different time scales, this is the fundamental of higher-order moment method.

(2) The first six Bernoulli polynomials Eqs. (17)–(22) have been found. These polynomials have the character: whens > 1.0, C2, C4 and C6 are negative numbers, but C3 andC5 are positive numbers.

(3) We obtain lattice Boltzmann model for the conservation law equation with the fourth-order accuracy of truncationerror. The dispersion term and dissipation term can be controlled to positive and negative.

This ideal and method of this paper can be spread into one-dimensional Euler equations and Navier–Stokes equations.Perhaps we could use multi-layer lattice Boltzmann model for the Euler equations and Navier–Stokes equations to yieldhigher-order accuracy of truncation error. These are our future works.

Acknowledgements

This work is Project 20092005 supported by Gradate Innovation Fund of Jilin University, the National Nature ScienceFoundation of China, (Grant No. 10072023, Grant No. 90305013), and the ChuangXin Foundation of Jilin University (No.2004CX041). We would like to thank Prof. Qian Yuehong, Prof. Wang Jianping, Liu Yanhong, Wang Huimin, Yan Bo, Shi Xiubo,Li Tingting, Chen Zhenlong, Chen Yuanjian, and Yin Xianli for their many helpful suggestions.


Appendix A. In order to obtain the conservation laws in different time scales, we need supply those partial differentials ontime ti to the equilibrium distribution function. Eqs. (11)–(16) are changed into the following forms:

@u@t0¼X

a�ea

@

@xf ð0Þa

� �; ðA:1Þ

@u@t1¼X

a�C2D

2f ð0Þa

h i; ðA:2Þ

@u@t2¼X

a�C3D

3f ð0Þa � 2C2D@

@t1f ð0Þa

� �; ðA:3Þ

@u@t3¼X

a�C4D

4f ð0Þa � 3C3D2 @

@t1f ð0Þa � 2C2D

@

@t2f ð0Þa � C2

@2

@t21

f ð0Þa

" #; ðA:4Þ

@u@t4¼X

a�C5D

5f ð0Þa � 4C4D3 @


2 @


@


@2

@t21

f ð0Þa

"�2C2

@2

@t1@t2f ð0Þa

#; ðA:5Þ

@u@t5¼X

a�C6D

6f ð0Þa � 5C5D4 @


3 @


2 @


@


2 @2

@t21

f ð0Þa

"

�6C3D@2

@t1@t2f ð0Þa � 2C2

@2

@t1@t3f ð0Þa � C3

@3

@t31

f ð0Þa � C2@2

@t22

f ð0Þa

#: ðA:6Þ

Appendix B. For the 3-bit model, those velocities are e0 = 0, e1 = c, e2 = �c. These equilibrium distribution functions can besolved by combining Eqs. (33)–(35), they are

f ð0Þ1 ¼ 12c2 ðm

0c þ p0Þ; ðB:1Þ

f ð0Þ2 ¼ 12c2 ð�m0c þ p0Þ; ðB:2Þ

f ð0Þ0 ¼ u� f ð0Þ1 � f ð0Þ2 : ðB:3Þ

For the 5-bit model, those velocities are ea = (0,c, �c,2c,�2c), where, a = 0,1, . . . ,4. These equilibrium distribution func-tions can be solved by combining Eqs. (33)–(37), they are

f ð0Þ1 ¼ 16c4 4m0c3 þ 4p0c2 � Q 0 � P0c

; ðB:4Þ

f ð0Þ2 ¼ 16c4 �4m0c3 þ 4p0c2 � Q 0 þ P0c

; ðB:5Þ

f ð0Þ5 ¼ 124c4 2P0c � 2m0c3 þ Q0 � p0c2

; ðB:6Þ

f ð0Þ4 ¼ 124c4 �2P0c þ 2m0c3 þ Q 0 � p0c2

; ðB:7Þ

f ð0Þ0 ¼ u� f ð0Þ1 � f ð0Þ2 � f ð0Þ3 � f ð0Þ4 : ðB:8Þ

For the 7-bit model, those velocities are ea = (0,c,�c,2c,�2c,3c,�3c), where, a = 0,1, . . . ,6. These equilibrium distributionfunctions can be solved by combining Eqs. (33)–(39), they are

f ð0Þ1 ¼ 1720c6 ð540p0 � 195Q 0c2 þ 15S0 þ 540m0c5 � 195P0c3 þ 15R0cÞ; ðB:9Þ

f ð0Þ2 ¼ 1720c6 ð540p0 � 195Q 0c2 þ 15S0 � 540m0c5 þ 195P0c3 � 15R0cÞ; ðB:10Þ

f ð0Þ3 ¼ 1720c6 ð�6S0 � 12R0c þ 60Q 0c2 þ 120P0c3 � 54p0c4 � 108m0c5Þ; ðB:11Þ

f ð0Þ4 ¼ 1720c6 ð�6S0 þ 12R0c þ 60Q 0c2 � 120P0c3 � 54p0c4 þ 108m0c5Þ; ðB:12Þ

f ð0Þ5 ¼ 1720c6 ðS

0 þ 3R0c � 5Q 0c2 � 15P0c3 þ 4p0c4 þ 12m0c5Þ; ðB:13Þ

f ð0Þ6 ¼ 1720c6 ðS

0 � 3R0c � 5Q 0c2 þ 15P0c3 þ 4p0c4 � 12m0c5Þ; ðB:14Þ

f ð0Þ0 ¼ u� f ð0Þ1 � f ð0Þ2 � f ð0Þ3 � f ð0Þ4 � f ð0Þ5 � f ð0Þ6 : ðB:15Þ


Appendix C. According to the definition of higher-order moments Eqs. (33)–(39), Eqs. (A.1)–(A.6) become the followingforms:

@u@t0þ @m0

@x¼ 0; ðC:1Þ

@u@t1þ C2

@

@x@m0

@t0þ @p

0

@x

� �¼ 0; ðC:2Þ

@u@t2þ C3

@

@x2@2m0

@t20

þ 3@2p0

@x@t0þ @

2P0

@x2

!¼ 0; ðC:3Þ

@u@t3þ C4

@

@x3@3m0

@t30

þ 6@3p0

@x@t20

þ 4@3P0

@x2@t0þ @

3Q0

@3x3

!þ 3C3

@

@xð @

2m0

@t0@t1þ @2p0

@x@t1Þ þ C2

@2u@t2

1

¼ 0; ðC:4Þ

@u@t4þ C5

@

@x4@4m0

@t40

þ 10@4p0

@x@t30

þ 10@4P0

@x2@t20

þ 5@4Q 0

@x3@t0þ @

4R0

@x4

!

þ 4C4@

@t1

@

@x2@2m0

@t20

þ 3@2p0

@x@t0þ @

2P0

@x2

!ðC:5Þ

þ 3C3@

@t2

@

@x@m0

@t0þ @p

0

@x

� �þ 2C2

@2u@t1@t2

¼ 0;

@u@t5þ C6

@

@x5@5m0

@t50

þ 15@5p0

@x@t40

þ 20@5P0

@x2@t30

þ 15@5Q 0

@x3@t20

þ 6@5R0

@x4@t0þ @

5S0

@x5

!

þ 5C5@

@t1

@

@x3@3m0

@t30

þ 6@3p0

@x@t20

þ 4@3P0

@x2@t0þ @

3Q0

@3x3

!

þ 4C4@

@t2

@

@x2@2m0

@t20

þ 3@2p0

@x@t0þ @

2P0

@x2

!þ 3C3

@

@t3

@

@x@m0

@t0þ @p

0

@x

� �ðC:6Þ

þ 6C4@2

@t21

@

@x@m0

@t0þ @p

0

@x

� �þ 2C2

@2u@t1@t3

þ C3@3u@t3

1

þ C2@2u@t2

2

¼ 0:

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A higher-order moment method of the lattice Boltzmann model for the conservation law equation

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