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A lattice Boltzmann model for reaction dynamical

systems with time delay

Yu Xiaomeia, Shi Baochanga,b,*

aState Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, PR China

bDepartment of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China

Abstract

Nonlinear reaction dynamical systems with time delay are investigated to evaluate the capability of the lattice Boltzmann

model for delay differential dynamical systems. The stiff delay systems, multi-delay systems and two-dimensional partial

delay differential equation are studied. Computation precise, efficiency, delay appearances are observed. The simulation

results indicate that the lattice Boltzmann method is an innovative and effective numerical method to simulate the nonlinear

delay differential dynamical systems.

? 2006 Elsevier Inc. All rights reserved.

Keywords: Nonlinear reaction dynamical systems; Time delay; Lattice Boltzmann method

1. Introduction

The lattice Boltzmann method (LBM) is an innovative numerical method based on kinetic theory to sim-

ulate various hydro-dynamical systems [1,2]. Although the LBM was developed only a decade ago, it has

shown large power to simulate various complex dynamical appearances [3–5]. Compared with conventional

numerical methods, it has advantages of simplicity, less computation, intrinsic parallel and so on. LBM

has been considered recently as a possible alternative explicit numerical scheme to conventional methods

for solving nonlinear macroscopic physical systems, in particular for Navier–Stokes flows. Consequently,

LBM has found applications in many areas of flow physics [6–8].

An important application of LBM is to solve a class of generalized reaction–diffusion systems, which

include ordinary differential dynamical systems, convection–diffusion, pure diffusion, Poisson equations. It

is more difficult than Navier–Stokes via LBM for the momentum flux is not conservative when reproduce

the desired form of nonlinear reaction–diffusion equations. Some studies using LBM to simulate nonlinear

reaction dynamical systems have been performed [9,10]. But seldom work has referred to the nonlinear

reaction–diffusion dynamical systems with time delay, which is more complex and can lead to surge and chaos.

0096-3003/$ - see front matter ? 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.02.020

*Corresponding author. Address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074,

PR China.

E-mail addresses: yuxiaomei_hust@126.com (X. Yu), sbchust@126.com (B. Shi).

Applied Mathematics and Computation 181 (2006) 958–965

www.elsevier.com/locate/amc

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Many conventional methods [11–14] have studied to solve the delay differential equations. But many methods

have restrict in delay terms and still lack of quantitative results. They may require the delay terms to be lin-

earity, monotone, small delay number and so on. Large amount of computation and the data conserved is one

aspect of existing problems. Especially, when the problem is stiff, more restrictions of the numerical method

are required. Moreover, the partial differential delay systems still lack deep work.

In this paper, we focus on the application of LBM to delay differential systems. We use our lattice

Boltmann model with adjustable parameter based on the without source assumption model [15], which has

more generalized applied area than standard lattice Boltzmann models, to simulate stiff delay, multi-delay,

partial delay systems, respectively. In Section 2, the lattice Boltzmann model with adjustable parameter is

given. Section 3 shows the numerical simulations for varied delay reaction dynamical systems.

2. The lattice Boltzmann model

Consider a DnQb (n dimensions in space and b discrete directions) model [2],a lattice with unit spacing is

used where each node has b nearest neighbors connected by b links. We just take D2Q9 model for example. It

is easy to development any DnQb model in same method. The particles velocity vectors are defined by

8

>

where c = Dx/Dt, Dx and Dt are the lattice grid space and time step, respectively; c is the particle speed. The

evolution equation of the density distribution function in the LBGK model reads

?

The model can get rid of the source assumption [15] existing in the other literature by add the1

delay terms also can be treat as source term in the simulations. where s is the dimensionless relaxation time,

and

h

xiq a þci?u

csis the sound speed. here c2

relaxation parameter which is used to reproduce ordinary delay system. As in the usual Chapman–Enskog

expansion, the fi(x,t) and feq

i

satisfy

X

X

X

and according the definition of Fi(x,t), it satisfiesP

oq

otþ u ? rq ¼ c2

2

ci¼

ð0;0Þ;

ðcos½ði ? 1Þp=2?;sin½ði ? 1Þp=2?Þc;

ðcos½ð2i ? 9Þp=4?;sin½ð2i ? 9Þp=4?Þ

i ¼ 0;

i ¼ 1;2;3;4;

i ¼ 5;6;7;8;

ffiffiffi

2

p

c;

>

:

<

fiðx þ ciDt;t þ DtÞ ? fiðx;tÞ ¼ ?1

s

fiðx;tÞ ? fðeqÞ

i

ðx;tÞ

?

þ DtFiðx;tÞ þ1

2Dt2oFiðx;tÞ

ot

.

ð2:1Þ

2Dt2oFiðx;tÞ

ot. Then

feq

i

¼

ð1 ? aÞq þ xiq a þci?u

sþ 0:5ðci?uÞ2

c2

sþ 0:5ðci?uÞ2

? 0:5juj2

c4

i

s

? 0:5juj2

c2

s

i

;

i ¼ 0;

i 6¼ 0;

c2

c4

s

c2

s

h

;

8

>

>

:

<

s¼c2

3, x0= 4/9, xi= 1/9(i = 1,2,3,4), xi= 1/36(i = 5,6,7,8), and a is an extra

i

fiðx;tÞ ¼

X

i

fðeqÞ

i

ðx;tÞ ¼ q;

ð2:2Þ

i

cifðeqÞ

i

ðx;tÞ ¼ qu;

ð2:3Þ

i

cicifðeqÞ

i

ðx;tÞ ¼ ac2

sqI þ quu

ð2:4Þ

iFi¼ F;P

iciFi¼s?0:5

sFu. Then the hydro-dynamic equa-

tion derived from Eq. (2.1) through the Chapman–Enskog analysis is

?

More than one lattice Boltzmann evolution equations must be used if the dynamical system is multi-

components

sDta s ?1

?

r2q þ F.

ð2:5Þ

fs

iðx þ ciDt;t þ DtÞ ? fs

iðx;tÞ ¼ ?1

ssðfs

iðx;tÞ ? fs;eq

i

ðx;tÞÞ þ DtFs

iðx;tÞ þ1

2Dt2oFs

iðx;tÞ

ot

;

ð2:6Þ

X. Yu, B. Shi / Applied Mathematics and Computation 181 (2006) 958–965

959

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where fs

tively.

Then the diffusion coefficient can be determined by

?

We can find through adjust the parameter a, we cannot only simulate the partial problems which are always

done in much work of the LBM, but also the ordinary differential dynamical systems, which cannot achieve

the goal without the adjustable parameter a in previous model.

i;fs;eq

i

is the distribution function and equilibrium distribution function of s component, respec-

D ¼ c2

sDta s ?1

2

?

.

3. Numerical simulations

In the following, we present some numerical tests to validate the lattice Boltzmann method for nonlinear

dynamical systems. The nonequilibrium extrapolation method [16] is adopted to deal with boundary condi-

tions in simulations. During the simulation, the double precision type is used for each variable. The simula-

tions are performed on a Pentinum 4 (2.4 G).

3.1. A stiff delay differential system

Apply the lattice Boltzmann method to the following delay differential equation [17]

y0ðtÞ ¼ ?1000yðtÞ þ 999:9yðt ? 1Þ;

yðtÞ ¼ e?ðn0tÞ;

where n0= 0.99905 · 10?4and the exact solution is yeðtÞ ¼ e?ðn0tÞ.

We can see that the stiff rate is rD= 1999. As we known, many numerical methods cannot be applied to stiff

problems for its extra requirement of step size and stability. The D1Q3 model is adopted to simulate the stiff

system by choosing a = 0, and the discrete velocities is ci= 0,1,?1 (i = 0,1,2) with the weight coefficients

xi= 2/3,1/6,1/6, respectively. The results are compared with [17] by explicit Euler, implicit Euler method.

From Table 1, we can see that though the LBE method used is an explicit method, it can achieve better results

than the implicit Euler method. That is to say, the lattice Boltzmann method is a more effective and valuable

numerical method when applied in stiff delay systems. The absolute error is defined as

t > 0;

ð3:1Þ

ð3:2Þ

t 6 0;

err ¼ jyðtÞ ? yeðtÞj;

where y(t) is the numerical solution and ye(t) is the exact solution. Figs. 1 and 2 tell that the stiff delay system

evolutes with delay time periodically. Once the evolution time is long enough, the delay effect would decrease

gradually. The results are in agreement with classical ones.

ð3:3Þ

3.2. A multiple delays system

As the delay system have more than one delay, the system would be more complex and harder to control.

The delays would strictly effect the performance of the systems. Many work focus on discussing the stability of

multiple delays systems in analytical method. The direct and simple numerical simulations is lack. A multiple

delays system shown below is simulated via lattice Boltzmann method

Table 1

The results of different computation method

Varied methodsStep size Iteration stepResult

LB

Explicit Euler

Implicit Euler

The exact solution

0.001

0.001

0.1

–

2.3 · 107

2.3 · 107

2.3 · 105

–

0.1004779

0.10025

0.10049

0.100478

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X. Yu, B. Shi / Applied Mathematics and Computation 181 (2006) 958–965

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y0

y0

1ðtÞ ¼ ?y1ðtÞy2ðt ? 1Þ þ y2ðt ? 10Þ;

2ðtÞ ¼ y1ðtÞy2ðt ? 1Þ ? y2ðtÞ

yðtÞ ¼ ½5;0:1?>;t 6 0.

We can see that there are discontinuous points existing in t = n. These points make the computation is harder

and lead to many numerical method instability and lower precision. when t = 40, the referred solution is given

?

We also use the D1Q3 to simulate the multiple delay systems by using two lattice Boltzmann evolution equa-

tions. The results at t = 40 are shown in Table 2. The computational results prove that LB method can solve

t P 0;

?

y1ð40Þ ¼ 0:0912491205663460;

y2ð40Þ ¼ 0:0202995003350707;

0123

t

456

0

1

2

3

4

5

6x 10

–10

err

Fig. 1. The absolute error between numerical solutions and exact solutions varying as time.

2.2085 2.2092.2095

t

2.212.2105

x 10

4

2.313

2.3135

2.314

2.3145

2.315

2.3155

2.316

2.3165x 10

–7

err

Fig. 2. The zoom portion of absolute error at long time with finer details.

Table 2

The results at t = 40 varying at different step size

Step size

y1(40)

y2(40) Computer time (clocks)

Dt = 0.01

Dt = 0.001

Dt = 0.0001

Dt = 0.00001

0.085415

0.090651

0.091189

0.091246

0.020742

0.020342

0.020304

0.020299

0.000006

0.016

0.047

0.422

X. Yu, B. Shi / Applied Mathematics and Computation 181 (2006) 958–965

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the multiple system accurately, simply and with small computation. Though the system has fluctuation, the

simulate results still have good precision with lattice Boltzmann method without interpolation. The appear-

ance of system varying of time shows in Fig. 3.

3.3. A two-dimensional partial reaction system with nonlinear source

A partial reaction system is given as

?

uðx;y;tÞ ¼ e?tsinðx þ yÞ;

The source term

ou

ot? D

o2u

ox2þo2u

ox2

?

¼ fðx;y;t;u;usÞ;

t 6 0.

x;y 2 ð0;1Þ;t 2 ð0;TÞ;

ð3:4Þ

ð3:5Þ

f ¼ uð1 ? u ? usÞ þ wð2p2? 2 þ w þ eswÞ;

where w = e?tsin(x + y) and the exact solution is u(x,y,t) = e?tsin(x + y).

Different conditions varying with diffusion coefficient and time step size at each time are simulated. The

absolute error at different time is defined as

err ¼ juði;j;tÞ ? ueði;j;tÞj.

The error at different time step and different diffusion coefficient, delay is shown in Figs. 4 and 5. When the

delay time increase, the difficult of computation has added and leads to the computation precise decrease.

ð3:6Þ

0

0.5

1

0

0.5

1

0

2

4

6

x 10

–4

x

y

err

0

0.5

1

0

0.5

1

0

1

2

x 10

–4

x

y

err

Fig. 4. The absolute error between numerical solution and exact solution at time step = 5000, 6000, respectively. s = 1, D = 0.1,

Dt = 0.001.

0 100200

t

300400

0

1

2

3

4

5

6

y1(t)

0 100 200

t

300 400

0

0.5

1

1.5

2

y2(t)

Fig. 3. Numerical solution of Eqs. (3.4), (3.5) for Dt = 0.01.

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X. Yu, B. Shi / Applied Mathematics and Computation 181 (2006) 958–965