lattice dependence of reaction-diffusion in lattice ... · we consider the inﬂuence of the...

Computer Physics Communications 129 (2000) 256–266www.elsevier.nl/locate/cpc

Lattice dependence of reaction-diffusion in latticeBoltzmann modeling

Ronald Blaak1, Peter M.A. Sloot∗University of Amsterdam, Faculty for Mathematics, Computer Science Physics and Astronomy, Section Computational Science,

Kruislaan 403, 1098 SJ Amsterdam, The Netherlands

Abstract

We consider the influence of the lattice symmetry and size of the lattice Boltzmann method on the behavior of a pure reactiondiffusion system. We show that the effect of the dispersion relation in the diffusion coefficient can be minimized, by tuning thefraction of rest particles and the relaxation parameter. For the reaction, we focus on the Selkov model and study the dynamicsof pattern formation due to the Turing Instability. For the chosen reaction parameters, however, no clear influence of the latticesymmetry is found. 2000 Elsevier Science B.V. All rights reserved.

PACS:47.11.+j; 47.54.+r

Keywords:Lattice-Boltzmann method; Reaction-diffusion; Lattice symmetry; Pattern formation

1. Introduction

In the last few years a lot of attention is paid to re-action diffusion systems. Experimental and numericalresults have shown that these in principle simple sys-tems show a wide variety of complex behaviors, suchas chemical waves, oscillations and non-homogeneouspatterns [1–3]. These systems are for instance formedby a number of chemical species which are dissolvedin some fluid and are able to chemically interact witheach other forming new compounds. The macroscopicequations which govern these phenomena, as well asthe general behavior of these systems, can also be ob-served in different models and or simulations, e.g.,population dynamics.

∗ Corresponding author. E-mail: [email protected] E-mail: [email protected].

The chemical species are allowed to react, whichgives rise to a dynamical process to form a steady stateor dynamically stable cycle. By tuning the reactionparameters, one is able to change this process suchthat the final outcome is a homogeneous state, astable pattern, formed by domains where differentchemically species are dominant, or an oscillatingstate.

In general such a reaction-diffusion system can bedescribed by the following set of coupled equations:

∂ρs

∂t−Ds∇2ρs =Rs, (1)

wheret is the time,∇2 the Laplacian operator withrespect to the spatial coordinatex, ρs(x, t) the massdensity andDs is the diffusion coefficient. The sub-script s is a label for the different chemical species.

0010-4655/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0010-4655(00)00112-0

R. Blaak, P.M.A. Sloot / Computer Physics Communications 129 (2000) 256–266 257

The right-hand side of the equation,Rs , represents thereaction term and will in general depend on the lo-cal densities of all chemical species and the reactionmechanism governing this system. In reality the reac-tion term should also depend on the temperature, spe-cially since reaction will produce energy which willlocally heat up the system. However, we will assumeour model to be a-thermal.

In order to model more realistic systems large sys-tem sizes are required. For that reason we adopt thelattice Boltzmann formalism to simulate such sys-tems [4], because this method is ideally suited for par-allel execution, thus supporting large system sizes [5].Moreover it is also able to deal with complicatedgeometries [6] and changing environments, e.g., ag-gregation processes [7].

In this article we will investigate how the systemsize and the symmetry of the underlying lattice will in-fluence the behavior of the system. If one is interestedin the steady state solution of a reaction-diffusion sys-tem only, this is in principle not necessary. Providedthe system size which is modeled is large enough, it isconceivable that the lattice symmetry is not too impor-tant any steady state could be obtained.

If one is interested in dynamical modeling of prob-lems like ground water contamination and bioreme-diation, however, the lattice symmetry might be im-portant. The reason is that at a local level and shorttime scales the lattice symmetry might influence thedynamical behavior of reaction diffusion processes. Inorder to investigate this influence, we have performeda series of simulations on the formation of Turing pat-terns [8] in the Selkov model [9]. The choice for thisparticular model is based on its simplicity and the factthat much is known about it.

The remaining of this article is organized as fol-lows. In Section 2 we explain the model we used tosimulate the reaction diffusion. In Section 3 we com-pare the predicted diffusion coefficient with the onefrom simulations. In Section 4 the Selkov model is dis-cussed in some more detail. In Section 5 we presentour results on the size and symmetry dependence ofthe simulations. In Section 6 we consider the influ-ence of the boundaries. In Section 7 we finish witha discussion of our results and suggestions for futureresearch.

2. Model

In order to model the reaction-diffusion equations(1), we will use the lattice-Boltzmann scheme. Theone-particle distribution of speciess at timet , positionEx and with velocityEei is denoted byfs(Ex, i, t). In caseof a hexagonal lattice the nearest neighbor vectors aregiven by

Eei =(0,0), i = 0,(cos(θi),sin(θi)), θi = (i − 1)π/3,

16 i 6 6,(2)

and in the case of a square lattice by

Eei =

(0,0), i = 0,(cos(θi),sin(θi)), θi = (i − 1)π/2,

16 i 6 4, (3)√2(cos(θi),sin(θi)), θi = (i − 5)π/2+ π/4,

56 i 6 8.

The lattice Boltzmann equation forfs(Ex, i, t) can bewritten as

fs(Ex + Eei, i, t + 1)− fs(Ex, i, t)=Ωs(Ex, i, t), (4)whereΩs(Ex, i, t) is the collision operator of speciess. This term consists of a reactive termΩRs and non-reactive termΩNRs . For the non-reactive part of thecollision we use the normal single relaxation timeBGK approximation:

ΩNRs (Ex, i, t)=−1

τs

(fs(Ex, i, t)− f eqs (Ex, i, t)

), (5)

where the equilibrium densityf eqs depends on the lo-cal densityρs and local velocityEu. In the normal lat-tice Boltzmann simulation this equilibrium distribu-tion for is given by

feqs (Ex, i, t)=ws,iρs

(1+ (Eei · Eu)

c2s+ (Eei · Eu)

2

2c4s− Eu

2

2c2s

),

(6)

wherecs in the speed of sound. The weightsws,i aredependent on the lattice symmetry, and result in

ws,i ={zs, i = 0,(1− zs)/6, 16 i 6 6 (7)

258 R. Blaak, P.M.A. Sloot / Computer Physics Communications 129 (2000) 256–266

for a hexagonal lattice, wherezs denotes the fractionof rest particles and can be different for differentspecies. For a square lattice we find

ws,i =zs, i = 0,(1− zs)/5, 16 i 6 4,(1− zs)/20, 56 i 6 8.

(8)

Since we are modeling a system where the chemicalspecies are dissolved in a fluid, the velocityEu, can bereplaced by the local velocity of the solvent, whichfor low concentrations will hardly be influenced by thereactants. In what follows, however, we will consider apure reaction diffusion system, and hence the averagemean flow will be zero, this leads to the reduction ofthe equilibrium distribution function to

feqs (Ex, i, t)=ws,iρs =ws,i

∑i

fs(Ex, i, t). (9)

For the reactive part of the collision term, we willfollow Dawson et al. [4], and assume thatRs thechange in density of speciess due to a reaction, will bedistributed over the different directions, proportionalto the weights of pure diffusion

ΩRs =ws,iRs . (10)The exact form the reaction termRs , still needs to

be specified. It depends on the set of reactions oneis interested in, and can be very complicated. In ourapplication we will assume that it only depends onthe local densities of the reacting species, howeverit should be noted that in reality this might alsodepend on the velocity dependent cross-sections forthe reaction.

If we assumef ≈ f eq+ �f (1), ∂/∂t ∝ �2, ∂/∂ Ex ∝�, andRs ∝ �2, where� is a small parameter, a Taylorseries offs(Ex + Eei, i, t) aroundfs(Ex, i, t) will lead tothe desired evolution equation (1) forρs [4], and thediffusion coefficientDs results in

Ds = 12(1− zs)(τs − 12

)hexagonal lattice, (11)

Ds = 35(1− zs)(τs − 12

)square lattice. (12)

In the case of pure diffusion, as we intend to use ini-tially, this fraction can be chosen arbitrary to meet ex-ternal constraints. If one is interested in the applicationincluding flow, these values are chosen to bezs = 1/2for the hexagonal andzs = 4/9 for the square lattice,in order to retrieve the Navier–Stokes equations. Note

that the relaxation parameter has to be chosen largerthan a half, otherwise unphysical behavior is obtained.

3. Diffusion coefficient

In order to measure the diffusion coefficient we usea non-reactive system in which we impose initiallya concentration wave of a sinusoidal shapeρ(Ex, t =0)= 1+� cos(Ek · Ex) for some wave vectorEk. This wavevector should however be chosen such that it fits on thelattice, e.g.,Ek = (2πi/Nxx̂,2πj/Nyŷ).

If this density profile is applied to the diffusionequation we find that it has an exponential decay givenby

ρ(Ex, t)= ρ(Ex,0)e−Dk2t . (13)In Fig. 1 we have plotted the measured diffusioncoefficient and its relative error as a function of therelaxation parameter for several wave vectors. In orderto measure the diffusion, the decay of the imposedamplitude as function of the time is fitted to anexponential. Although the simulations are performedon a 256× 256 grid, this is not strictly necessary forall wave vectors. The only requirement for a givenwave vector is that it should fit an integer number oftimes on the lattice, hence, depending on the wavevector, smaller lattices can be used. The wave vectorsshown here areEk = (i2π/256, j2π/256) with i, j =0,1,2,4,8.

It can be seen that the behavior of the measured dif-fusion coefficients for different wave vectors is self-similar an can, for larger relaxation parameters, bemapped on each other. However, it is clear that forτ > 5 the deviations from the predicted diffusion co-efficient become too large for out purposes. The rel-ative error of indicates that also for small relaxationparameters the measured and predicted value disagreeincreasingly for larger wave vectors. In first approxi-mation, however, the curves share a common value forthe relaxation parameter is optimal.

The origin of the deviation of the measured diffu-sion from the predicted one, lies in the decay of theamplitude of the imposed density profile. For smallvalues ofτ & 0.5 the relaxation goes fast, which leadto numerical instabilities in simulations causing initialoscillations, rather than a perfect exponential decay.


(a) (b)

Fig. 1. (a) The measured diffusion coefficientD as function of the relaxation parameterτ . The straight line is the theoretical value. (b) Therelative error in the measured diffusion coefficient as a function of the relaxation parameter for smallτ . The measurements are done on a256× 256 square lattice with the fraction of rest particlesz is fixed to 4/9. The labeling of a curve(i, j) correspond to the wave vector2π256(i, j).

The diffusion coefficient, however, will be given rea-sonable accurate. For largeτ the measured and pre-dicted value of the diffusion start to disagree. Due tothe numerical scheme, the amplitude is not decayingexponentially all the time, but can grow for small pe-riods.

The discrepancy between the measured and pre-dicted value of the diffusion coefficient, will put someconstraints on the simulation parameters used. De-pending on the range of interest, there are a few simplesolutions, of which the simplest would be taking a big-ger lattice size. To avoid both described effects in thedecay, one should take the relaxation parameter of theorder unity. In order to access larger diffusion coeffi-cients, one could use multiple time steps.

For pure reaction diffusion however, there is alsoanother option. Since the equilibrium distribution doesnot depend on the local velocity, one can chosedifferent weights, allowing to set a different value forthe diffusion. This can only be done, because theywere fixed by the requirement to retrieve the Navier–Stokes equations. In Fig. 2 the relative error of thediffusion is plotted as a function of the fraction of fixedparticles, forτ = 1 and several wave vectors. Alsohere the discrepancy increases for larger wave vectors.

Fig. 2. The relative error in the measured diffusion coefficientas a function of the fraction of rest particlesz. Different curvescorrespond to different wave vectors in magnitude and direction ona 256×256 square lattice, with the relaxation parameterτ = 1. Thelabeling of a curve(i, j) correspond to the wave vector2π256(i, j).

For different values of the fraction of rest particlesz, figures similar to Fig. 1 are obtained. However,this leads to a shift of the “optimal” value for therelaxation parameter. In case of the square grid forz=3/9, 4/9 and 5/9 this value is given by respectivelyτ ≈ 1.07, 1.00 and 0.96. Since smallerz and largerτ


both increase the diffusion coefficient (11), this can beused to simulate a diffusion coefficient which is lesssensitive to a dispersion relation.

Although the theoretical diffusion coefficient, whichis obtained from the Chapmann–Enskog expansion, isindependent of the wave vector, this is obviously nottrue. In order to eliminate this effect as much as pos-sible the values of the relaxation parameter and thefraction of rest particles should be chosen appropri-ately. For the fraction of rest particles, we keep thevalues as used for normal LBM simulations. The mainreason for this choice is that in future applicationsthe incorporation of low flow fields is needed and wedon not pursue the direction of pure reaction-diffusion,where this value could possibly be optimized. Thesame is true for the relaxation parameter, which nor-mally would be close to unity. However in order theobtain a sufficient range of diffusion constants, we al-low it to be chosen in the range(0.55,2.0).

The existence of the dispersion relation, however,will in general affect the simulations. The deviation ofthe diffusion coefficient from the predicted one, willshow up mainly in the larger wave vectors and hencesmaller length scales. Depending on the applicationthis will influence the outcome of the simulation. Inour case, where we will focus on the formation ofpatterns, this could for instance influence the shape ofthe spots that appear in the pattern. The relevant lengthscale of the pattern is an order of magnitude larger andwill be influenced less.

4. Selkov

The Selkov model consists of a system of threecoupled chemical reactions, involving four species [9]

Ak1

k−1X, (14a)

X+ 2Y k2

k−2

3Y, (14b)

Yk3

k−3B. (14c)

It is assumed that the densities ofA andB are fixedby external sources. This results in only two evolutionequations, determined by the densities and the reactionrateski

∂ρX

∂t−DX∇2ρX=RX = k1ρA − k−1ρX − k2ρXρ2Y + k−2ρ3Y , (15a)

∂ρY

∂t−DY∇2ρY=RY = k−3ρB − k3ρY + k2ρXρ2Y − k−2ρ3Y . (15b)By solving the equationsRX = RY = 0, we find

that there are at most three fixed points. In order todetermine the linear stability of the fixed point withdensityρ∗X andρ∗Y , we study the exponential decay ofa homogeneous distortionEρ∗ + E� exp(λt). This givesa simple eigenvalue problem∂t Eρ = L Eρ, whereLij =∂Ri/∂ρj and results in the eigenvalue equation

λ2− Tr(L)λ+ |L| = 0. (16)For a stable solution we require Re(λ±) < 0. In thecase of two complex solutions this means thatL11+L22< 0, in the case of two real solutions there is anadditional constraint, the determinant of the matrixLshould be positive, otherwise there would exist both anegative and positive eigenvalue, hence, a stable andunstable mode. In the last case the simulation will godirectly to the stable fixed point, while in the first casethe density will spiral to that point.

If the ratio of the diffusion coefficientsDX/DYbecomes larger than a critical value, the Selkov modeldevelops a Turing instability. The fixed point becomeslinearly unstable against perturbations of the formδ Eρ =∑Eξ EAEξeıEξ ·Ex . For a complete analysis we referto [10], while we give here only the main results. Thecritical ratio of the diffusion coefficients is given by

DX

DY= LXXLYY − 2LXYLYX

L2YY

+ 2√LXYLYX(LXYLYX −LXXLYY )

L2YY

. (17)

If the ratio of the diffusion coefficients is larger thanthis critical ratio the Turing instability is guaranteedfor those wave vectorsξ which satisfy|ξ−| 6 |ξ | 6|ξ+| where|ξ2|± is given by|ξ2|± = DXLYY +DYLXX

2DXDY |L|

±√(DXLYY +DYLXX)2− 4|L|DXDY

2DXDY |L| .(18)


Fig. 3. Enlargement of the path followed by the average densitiesin time. This is the result of a simulation on a 256× 256 squaregrid where the spiraling behavior is ended after approximately 20thousand time steps.

5. Results

We have performed simulation of various latticesizes. We prepare the initial densities of both simulatedspecies in the Selkov model, by an average density,corresponding to the homogeneous solution of thereaction equations, with a random noise of aboutone percent. We have chosen he reaction parametersk1ρA = 2.65667× 10−3, k−1 = 6.65× 10−4, k2 =k−2 = 1.5 × 10−2, k−3ρB = 5.313334× 10−4 and

k3 = 6.65× 10−3 [4]. The diffusion coefficients areset toDX = 0.2914 andDY = 0.017, which ensuresthat the ratioDX/DY = 17.1412 is above the criticalvalue 16.2121. Under these conditions most of thepatterns formed in simulations with sizes 32m× 32nlead to the formation of spots in a hexagonal typearrangement. If the system size is to small, e.g., 32×32, homogeneous density distributions are found, andby narrowing only one of the lattice dimensions apattern of stripes is formed. In principle other patterns,e.g., arrangements with square symmetry, could beobtained by, initializing the densities of both specieswith specific modulations, however, these are easilydistorted.

In Fig. 3 an enlargement of the path of the averagedensities followed by both species in time, is shownfor a system on a square lattice with size 256× 256.It follows an inward spiral towards the stable fixedpoint of the reaction equations, as can be expectedon the basis of the stability analysis. However, whenthe fixed point is approached after approximately 20thousand time steps, the solution starts to run awayfrom this initial goal. What has happened is that thereaction equations initially hold not only locally butalso on average for the whole system, leading to thespiral path. In the neighborhood of the fixed pointhowever, the stable fluctuations are probed, forming

(a) (b)

Fig. 4. The integrated Fourier transform of the density profile of speciesX as function of the simulation time. The three time scales can beobserved, (a) spiraling to the fixed point (. 2×104 timesteps), (b) exponential growth of stable density modulations (. 2×105 timesteps) andthe diffusion of spots in order to form a regular pattern.


Fig. 5. Density profiles of speciesX in a 256× 256 simulation on a square lattice. The snapshots are taken after, from left to right, 0, 2× 104,5× 104, 105, 2× 105, 4× 105, 6× 105, 8× 105, and 106 timesteps.

the beginning of patterns. This creates gradients whichenter the reaction equations and cause them to be nolonger valid globally.

In the total simulation there are three time scalespresent, as can be seen in Fig. 4, where the integratedFourier transform is plotted as a function of thesimulation time. Initially there are oscillations, whichcorrespond to the inwards spiral to the stable fixed

point of the reaction equations, during which the initialrandom noise will diffuse and disappear and will takeabout 20–30 thousand time steps.

In the next stage the stable modes are probed froman almost homogeneous system, leading to a rapidgrow and to the formation of a clear pattern (Fig. 5),which is reached after about 200 thousand time steps.The initial pattern which is formed for sufficiently


Fig. 6. Bond-orientational order of neighboring maxima of speciesX as function of time.Q4 andQ2 remain finite, indicating that notrue six-fold symmetry is obtained.

large system, is formed a number of spots, with onlysome local order. In the last stage the spots start torearrange themselves, forming a regular pattern, in adiffusion like process.

The characterization of the pattern and its evolutionwill depend on the underlying lattice of the simulation,because only a limited range of wave vectors arestable and can lead to pattern formation. From thiscontinuous range, however, only few remain, i.e. thosewhich will fit on the lattice used in the simulation.Their number is finite and depends on both thedimensions of the lattice and its symmetry.

In order to examine this we analyze the patterns insome more detail. As function of time we determinethe locations of the extrema of the density profiles ofboth species. For speciesX we consider the maxima,while for speciesY we take the minima, whichare found at approximately the same locations inequilibrium. The shape of the spots is ellipsoidal, andin order to characterize them we fit the local densitydistribution within a distance of 5 lattice units by asecond order polynomial. By doing so we find thevalue of the extremum, its off-lattice location, and thetwo radii of the ellipsoid. The outcomes are relativelyinsensitive to the size of the neighborhood if takenbetween 2 and 10 lattice units.

The number of spots shortly after the initializationis fairly high, and also the number of spots forspeciesY is larger. After about 5000 steps the numberhas decreased already to a value close to the final

Fig. 7. The radii, describing the ellipsoidal shape of the maximaof speciesX, as function of time. The magnitude is a measurefor the steepness of the maximum, the relative difference for theeccentricity. The circles and squares correspond to the averagemaximum and minimum radius, respectively.

one. The relative difference between the magnitudeof the extrema and the average density, however, isof the order 10−3 and 10−4 for speciesX and Y ,respectively, and will continue to grow exponentiallyduring the second time-scale of the process.

In order to analyze the local order of the spots wedetermine the location of neighboring spots using theVoronoi construction. This allows us to calculate theaverage distance between neighboring spots and bond-orientational order parameters

Ql ={〈cos2(lφ)〉 + 〈sin2(lφ)〉}1/2, (19)

whereφ is the angle made by the line connecting twospots with the positivex-direction (Fig. 6).

As can be seen in the evolution of the pattern(Fig. 5), some spots merge after collision, while otherssplit. As a consequence the average distance betweenspots is only a useful property near equilibrium, whenthis number remains fixed. The bond-orientationalorder parameters are less sensitive to the numberof spots. Near equilibriumQ6 goes to unity, as isnecessary for a hexagonal pattern of spots. However,true six-fold symmetry is not obtained, becauseQ4andQ2 do not go to zero, but remain finite, indicatingthat there is still a preferred direction.

This is also made clear by the two radii describ-ing the ellipsoidal shape (Fig. 7). They differ aboutfifteen percent, indicating that a true six-fold symme-


try is not reached, because in that case it should becircular. These results are in agreement with the timeevolution of a 256× 296 system on a hexagonal lat-tice, which in real space has the same dimensions,and where relaxation parameters are chosen such thatthe theoretical diffusion coefficients in both simula-tions are identical. Although a complete comparisonis not possible, due to the different random initializa-tion in both cases, it does show that the overall behav-ior is similar, as it should. The distance between theextremes of the densities and bond-orientational orderparameters agree within the accuracy of the computa-tions.

This similarity is to be expected, since the macro-scopic parameters describing the pattern are deter-mined by the physical size of the system, which in bothcase is almost identical. If there is any influence of thelattice symmetry, it should be visible for small lengthscales. However, the height of the maximum densityof speciesX is 1.374 and 1.372 on the square respec-tively hexagonal lattice, for speciesY this is 0.265 and0.270. The shape of the spots on the hexagonal latticeis slightly more ellipsoidal.

In conclusion, in the final equilibrium state onlyminor differences exist due to the lattice symme-try. The same is observed in other smaller systemsizes. This does not mean that there is no influenceof the lattice symmetry on the equilibrium situation,but that the size of the spots in units of lattice spac-ing is sufficiently large. It is obvious that on de-creasing the size of the spots, the discrepancies be-tween the two lattice symmetries become more vis-ible. If for the present simulations a lattice symme-try dependence does exist, it could only be foundin the evolution of the pattern formation. However,since the initial density distributions also influencethis evolution, it is not possible to confirm or rule outthe existence of a dependence of the underlying lat-tice.

6. Boundaries

The results shown in the previous section are allobtained in simulations using periodic boundary con-ditions. Periodic boundary conditions are of course awidely used trick in computer simulations to simulatesmall system sizes, and never the less retrieving infor-

mation about much bigger systems. The trick howeverdoes not always give the desired result, as is the casehere.

As we are interested in pattern formation, and wantto obtain information about macroscopic systems, theperiodic boundary conditions work here against us.There are two reasons, first of all the lattice size incombination with the lattice symmetry will only allowfor certain length scales to be probed, i.e. those whichfit nicely on a square lattice are(i, j), where 06i 6M and 06 j 6M are arbitrary integers andMandN denote the lattice dimensions. The one withthe largest positive growth rate in the infinite system,will in general not be among those, but can only beapproximated.

There is, however, an additional problem comingfrom the periodic boundary conditions. As the Selkovmodel experiences a driving force to form a reg-ular pattern, it encounters the boundary conditions,which impose constraints on the pattern. To clarifythis, suppose that in an infinite system the optimalpattern would have stripes and the distance betweentwo maxima would be 64. If the oscillation wouldbe in thex-direction this would fit nicely on both a64× 32 and a 128× 32 lattice. If thex-dimensionof the lattice, however, would be anywhere in be-tween 64 and 128, it does not fit exactly. Therefore,although the preferred length scale is present, dueto the periodic boundary conditions, it can not beprobed.

There are two types of boundaries we can imposewhich differ from the periodic ones. The one we willpartially deal with here is walls, the other would beopen boundaries. The former is important, since inmost applications some impenetrable parts will bepresent, such as container walls or solid objects. Thelater could be used to avoid the coupling between thewave vectors and the lattice size.

The introduction of solid walls to the system,however, will have a different effect on the system. Itwill allow for all length scales present in the lattice,but the solid walls might act as a source of attractiveforce. In Fig. 8, the pattern evolution is shown for a256× 256 system on a square lattice, where we usedbounce back on the links. The dynamical behavior inestablishing a regular pattern is slower, as can be seensince the pattern is not yet stabilized after 106 timesteps, while the one with periodic boundaries (Fig. 5),


Fig. 8. Density profiles of speciesX in a 256× 256 simulation on a square lattice with solid walls. The snapshots are taken after, from left toright, 0, 2× 104, 5× 104, 105, 2× 105, 4× 105, 6× 105, 8× 105, and 106 timesteps. Note the defect in the last four time-shots, which is notobserved using periodic boundaries, that is slowly diffusing out of the system.

already is after 6× 105. Moreover, the last three time-shots show a defect, that is slowly diffusing out ofthe system, while such a defect is not stable usingperiodic boundaries. Although there is no constrainedto the length scales, the shape of the central spots, ison average still ellipsoidal and the walls seem to attractthe spots.

7. Conclusion

We have performed reaction diffusion simulationsin two dimensions. We used the Lattice Boltzmannmethod on various lattice sizes and two lattice sym-metries, square and hexagonal. Depending on the ap-plication, care should be taken in the construction of


such a model, because size and symmetry of the lat-tice could influence the outcomes seriously.

By focusing on the influence of the underlyinglattice on the formation of Turing patterns in theSelkov model, it is obvious that some finite sizeeffects are encountered. The equilibrium results do notdepend much on the symmetry of the lattice. If oneis interested in a dynamical modeling of the system,this influence might still be present in the patternevolution. However, in our simulations, where thedimension of the spot is of the order of 10 lattice units,no clear signature of the lattice symmetry is found.

One could of course adjust the simulation parame-ters in order to obtain smaller spot sizes, by decreas-ing the diffusion coefficients of both species. However,as a result the relaxation parameters would come tooclose to 0.5.

We indicated some preliminary results of simula-tions with boundary conditions other than periodicones, namely solid walls. The evolution itself takeslonger times to reach a truly equilibrium state. Theremoval of the periodic boundary conditions, releasesthe constraints on the patterns that can be formed.

However, it also means that the “force” it exerts onthe system to form a regular pattern is lost. Besides,also these boundaries influence the patterns, albeit in adifferent way.

References

[1] R. Kapral, A. Lawniczak, P. Masiar, Phys. Rev. Lett. 66 (1991)2539.

[2] J. Dethier, F. Baras, M.M. Mansour, Europhys. Lett. 42 (1998)19.

[3] Q. Ouyang, R. Li, G. Li, H.L. Swinney, J. Chem. Phys. 102(1995) 2551.

[4] S. Ponce Dawson, S. Chen, G.D. Doolen, J. Chem. Phys. 98(1993) 1514.

[5] D. Kandhai et al., Comput. Phys. Commun. 111 (1998) 14.[6] A. Koponen et al., Phys. Rev. Lett. 80 (1998) 716.[7] J.A. Kaandorp, C.P. Lowe, D. Frenkel, P.M.A. Sloot, Phys.

Rev. Lett. 77 (1996) 2328.[8] A.M. Turing, Philos. Trans. Roy. Soc. London, Ser. B 237

(1952) 37.[9] E.E. Selkov, Eur. J. Biochem. 4 (1968) 79.

[10] J.D. Murray, Mathematical Biology, 2nd edn. (Springer,Berlin, 1993).

lattice dependence of reaction-diffusion in lattice ... · we consider the inﬂuence of the...

Documents