Scalings and singular limits on the basis of the D1P2 model
2.1 Hyperbolic versus parabolic scaling
Hyperbolic scaling for balance and conservation laws
The lattice-Boltzmann equation in hyperbolic scaling is generally given by
∂tf+s∂xf =−ǫ−1ω
f−Ef
+qw, (2.1)
where ǫ denotes the scaling parameter and q is a prescribed source. Assuming the D1P2 model as the underlying discrete velocity set, equation (2.1) changes concretely into the system
∂tf1−∂xf1 = −ǫ−1ω
f1−(Ef)1
+12q , (2.2)
∂tf2+∂xf2 = −ǫ−1ω
f2−(Ef)2
+12q . (2.3)
The component f1 is the density of the particles moving in left direction, while f2 represents those particles heading to the right. Note that the equations are coupled via the equilibriumEf.
It turns out, that this system can be connected to the following hyperbolic conser-vation law1 with a continuously differentiableflux function2 f
∂tv+∂x f(v)
=q , (2.4)
if the equilibrium operator E is suitably chosen. In order to show this, let us introduce the mass moment u, which is physically the total mass density, and the first momentφ, whose meaning becomes more clear before long.
u:=hf,1i=f1+f2, φ:=hf,si=f2−f1 . (2.5) The mass moment occurs in the equilibrium operator E that we define3 by
Ef :=hf,1iw+f hf,1i
sw =uw+f(u)sw. (2.6) Since the moments of the equilibrium Ef are given by
hEf,1i =u=hf,1i, hEf,si=f(u)
we see that E conserves the mass moment. In contrast, the first moment is not preserved4. Let us now pass over to the equivalent moment system. This is ob-tained by performing an algebraic transformation of the lattice-Boltzmann system
1If the sourceqvanishes, we speak of aproperconservation law, otherwise the termbalance law is also used.
2Two well-known examples are the linear advection equation withf(v) =av (aconstant) and Burgers’ equation withf(v) = 12v2.
3Observe that (2.6) is the most general equilibrium in the case of the D1P2 model which depends only on the mass momentuand conserves it also.
4Otherwise this would implyφ=f(u); but this means that the first momentφwould be entirely determined byu. Since the variablesu, φare equivalent to the populationsf1,f2and since the latter ones are independent of each other, this would be a contradiction. However, the fact that the first moment of the equilibrium yields the flux anticipates the close relation between the first moment φand the flux, see equation (2.9).
2.1. Hyperbolic versus parabolic scaling 73
by expressingf1,f2 in terms of u, φ. Adding equation (2.2) and (2.3), i.e. applying h·,1i to (2.1), yields the evolution equation for u:
∂thf,1i+∂xhsf,1i= ǫ−1ωh
hEf,1i − hf,1ii
+qhw,1i
⇔ ∂tu+∂xφ= ǫ−1ω[u−u] +q=q .
Likewise, by subtracting (2.2) from (2.3), i.e. applyingh·,si to (2.1), the evolution equation for φis found:
∂thf,si+∂xhsf,si= ǫ−1ωh
hEf,si − hf,sii +q
z }| {=0
hw,si
⇔ ∂tφ+∂xu=ǫ−1ωh
f(u)−φi So we obtain the subsequent equivalent moment system
∂tu+∂xφ = q , (2.7)
∂tφ+∂xu = −ǫ−1ω
φ−f(u)
. (2.8)
Observe that this system corresponds just to therelaxation formulation5 pertaining to (2.4), that has been introduced by Jin and Xin in [29]. Solving the second equation (2.8) for φ
φ=f(u)−ǫτ ∂xu+∂tφ
(2.9) where we have setτ =ω−1 and plugging this into the first equation (2.7), produces finally
∂tu+f′(u)∂xu=q+ǫτ ∂t∂xφ+∂x2u
. (2.10)
It is possible to eliminate φ, such that we obtain a closed equation for u. Dif-ferentiating (2.7) with respect to t yields an expression for ∂t∂xφ. So we end up with
∂tu+f′(u)∂xu=q+ǫτ −∂2tu+∂tq+∂x2u ,
⇔ ǫτ ∂t2u+∂tu+f′(u)∂xu−ǫτ ∂x2u=q+ǫτ ∂tq . (2.11) Thus the mass momentusatisfies the balance law (2.4) if we formally setǫto zero.
In fact,u is expected to converge to the solution vof (2.4) under the assumptions, that φand its derivatives remain bounded forǫ↓0 and that the balance law (2.4) isstable under small perturbations.
In general, equation (2.11) determinesuin a unique way if supplemented by initial and boundary conditions. Onceu is known, φcan be computed by means of (2.9).
Differently from u, the first moment φ is determined as solution of an ordinary differential equation depending on the space variable x as parameter. Equation
5The namerelaxationis motivated by the observation, that the quantityφrelaxesin the space-homogeneous case towards the flux. This means that it is a solution of the ODE ˙φ+ǫ−1ωφ=f(u).
Starting from the balance law (2.4), as Jin and Xin did it,φappears as an artificially introduced variable. Our derivation of the relaxation system shows, that it is equivalent to a kinetic system, where we understand by akinetic systema PDE system that is formally similar to the Boltzmann equation (like the lattice-Boltzmann equation). In the kinetic frameworkφis interpreted as a first moment.
(2.9) assigns also a clear physical meaning to φ, because it approximates the flux f(u). Let us point out, that equation (2.8) (if multiplied byǫ) and equation (2.11) represent singularly perturbed differential equations, since the (highest) occurring derivatives are preceded by the scaling factor ǫ.
Equilibrium with two conserved moments
In order to illustrate how the equilibrium influences the target equation, let us consider the lattice-Boltzmann equation with another equilibrium:
Ef =hf,1iw+hf,sisw =uw+φsw. (2.12) In contrast to (2.6) this equilibrium conserves both moments since we have
hEf,1i=u=hf,1i, hEf,si=φ=hf,si.
Due to this circumstance the ǫ-dependence disappears, when we transform the lattice-Boltzmann equation into the equivalent moment system
∂tu+∂xφ = q , (2.13)
∂tφ+∂xu = 0. (2.14)
It should be noticed that also this system decouples, resulting in a separate wave equation for u andφ. In order check this, (2.14) is differentiated with respect to x and solved for ∂x∂tφ. Then it is inserted into (2.13) being also differentiated with respect to x. Similarly, the equation forφ is found.
∂t2u−∂x2u = ∂tq
∂t2φ−∂x2φ = −∂xq
The D1P2 model is too restricted to admit an adjustable wave speed. In this regard the D1P3 model accords more flexibility which is illuminated briefly by the subsequent excursion.
Equilibrium with two conserved moments for the D1P3 model
Let us shortly digress from the D1P2 model to consider an equilibrium analogous to (2.12) for the D1P3 model already discussed in 1.1.2. Recall thatsandware now given by
s= (−1,0,1)⊤, w=`1
2θ,θ−1θ ,2θ1´⊤
,
entailing s2 6=1and hw,s2i= 1θ. The moments u, φare defined as above; additionally a second order moment
ψ:=hf,s2−θ−1i
is introduced. An equilibrium preservinguandφis obviously given by Ef=hf,1iw+hf,siθsw=uw+φsw.
Applying the moment operators to the lattice-Boltzmann equation and observing hsf,si = hf,s2i=hf,s2−θ−1i+θ−1hf,1i=ψ+θ−1u , hsf,s2−θ−1i = hf,s3−sθ−1i= (1−θ−1)hf,si= (1−θ−1)φ ,
hEf,si = uhw,s2−θ−1i
| {z }
=0
+θφhsw,s2−θ−1i
| {z }
(1−θ−1)hw,si=0
= 0,
2.1. Hyperbolic versus parabolic scaling 75
we find the following equivalent moment system:
∂tu+∂xφ = q
∂tφ+1θ∂xu+∂xψ = 0
∂tψ+1θ∂xφ = −ǫ−1ωψ 9>
=
>; ⇔ 8>
<
>:
∂tu+∂xφ = q
∂tφ+θ−1∂xu = ǫτ ∂x`
∂tψ+ (1−1θ)∂xφ´ ψ = −ǫτ`
∂tψ+ (1−1θ)∂xφ´ The system to the right arises from the left one by solving the third equation forψand substituting the obtained expression into the second equation. A comparison of this system with the equations (2.13), (2.14) leads to the following observations:
• Since the second momentψis not conserved by the equilibrium, the right hand side of the evolution equation forψ(left below) does not vanish. Solving forψ, the evolution equation forφcan be written with a “source term” (right center) in contrast to (2.14). However the right hand side appears with the scaling factorǫ, such that it becomes of order O(ǫ) for well behaved bounded solutions. Therefore the mass momentu and the first momentφof the D1P3 model should converge forǫ↓0 to solutions of the wave equations∂t2v−θ−1∂x2v=∂tq and∂t2v−θ−1∂x2v=−∂xqrespectively.
• There is another point to be learned from this example. In 1.1.2 we have already seen how the D1P3 lattice-Boltzmann equation can be connected to a scalar conservation law (using the hyperbolic scaling as well). Amazingly, neither the algorithmic parameter τ (collision time) nor θ (weight parameter) interferes in the limit ǫ ↓ 0, i.e. in the target equation.
In contrast,θ is lent a physical meaning now: The second equation differs from (2.14) by the factor θ−1 in front of ∂xu. This shows, that the weight parameter θ determines the propagation speed of the waves. In chapter 3, we will see in the context of the parabolic scaling that the diffusivity is tuned byτ.
Parabolic scaling and viscous balance laws
Let us now consider the lattice-Boltzmann equation in parabolic scaling
∂tf+ǫ−1s∂xf =−ǫ−2ω
f−Ef
+qw. (2.15)
To recognize the impact of the scaling with regard to the target equation, we repeat the transformation into the equivalent moment system. For this we define the mass momentu and the first moment φanalogously to the first paragraph:
u:=hf,1i=f1+f2, φ:=hf, ǫ−1si= 1ǫhf,si= 1ǫ(f2−f1).
In contrast to (2.5), the first moment is rescaled6 by the factorǫ−1, as the quantity hf,si is deemed to be of magnitude O(ǫ). The equilibrium operator E is chosen as in (2.6) with the difference that the scaling factor ǫoccurs in front of the f-term
Ef :=hf,1iw+ǫf hf,1i
sw=uw+ǫf(u)sw.
Evidently, this equilibrium operator conserves also the mass moment. So u is ex-pected to play once more the favored role. Indeed, we will see thatu approximates the solution of the viscousbalance law this time:
∂tv+∂x f(v)
−ν∂x2v=q . (2.16)
6If we associate the populations f1,f2 with particles moving to the left or right respectively, then the hyperbolic and parabolic scaling distinguish each other by the following circumstance: In the case of the hyperbolic scaling the speed of the particles is constant 1 independently ofǫ. In opposition, the parabolic scaling requires, that the speed of the particles grows asǫtends to zero (inverse proportionality). If we assume that the first moment should be computed with respect to the (non-normalized) velocity (this seems physically more reasonable), then it is consequent to define the first moment as we have done it. However, a real justification of this definition is only obtained by a reasonable outcome.
We derive the equivalent moment system (evolution system for u and φ) by the same procedure as above. Applyingh·,1i to (2.15) results in the evolution equation foru:
∂thf,1i+ǫ−1∂xhsf,1i= ǫ−2ωh
hEf,1i − hf,1ii
+ghw,1i
⇔ ∂tu+∂xφ= ǫ−2ω[u−u] +q=q .
Analogically, the application of 1ǫh·,sito (2.15) leads to the evolution equation forφ:
ǫ−1∂thf,si+ǫ−2∂xhsf,si= ǫ−3ωh
hEf,si − hf,sii
+qhw,si
∂tφ+ǫ−2∂xhf,s2i=ǫ−3ωh
ǫf(u)−ǫφi
+ 0 = ǫ−2ωh
f(u)−φi . So we arrive at the equivalent moment system
∂tu+∂xφ = q , (2.17)
∂tφ+ǫ−2∂xu = −ǫ−2ω
φ−f(u)
. (2.18)
By solving the above equation for φ
φ=f(u)−τ ∂xu−ǫ2τ ∂tφ (2.19) and setting τ :=ω−1, (2.17) and (2.18) can be combined to
∂tu+f′(u)∂xu−τ ∂2xu=q+ǫ2τ ∂t∂xφ . (2.20) Identifying τ with the macroscopic parameterν (viscosity/diffusivity) we see, that the mass moment u satisfies a viscous balance law (2.16) apart from an additional source term. This term is however proportional to ǫ2 and should produce only a relatively small perturbation of (2.16). In analogy to equation (2.11) we obtain the following closed equation in u
ǫ2τ ∂t2u+∂tu+f′(u)∂xu−τ ∂x2u=q+ǫ2τ ∂tq . (2.21)
hyperbolic scaling parabolic scaling moments u=hf,1i, φ=hf,si u=hf,1i, φ= 1ǫhf,si equilibrium E(u) =uw+f(u)sw E(u) =uw+ǫf(u)sw LB equation ∂tf+s∂xf =−ǫ−1ω[f−Ef] ∂tf+ǫ−1s∂xf =−ǫ−2ω[f−Ef]
target equation ∂tv+∂x f(v)
= 0 ∂tv+∂x f(v)
−ν∂x2v= 0
Table 2.1: Synoptic summary comparing both of the scalings. The mass moment conserving equilibrium gives rise to a scalar target equation. The names of the scalings are borrowed from the classification of the target equations, that come out finally (particularly in this example).
In more complicated examples, however, the type of target equation/system might be of hybrid nature. Nevertheless, the parabolic scaling generally produces additional dissipative terms.
Therefore it is often referred to as thediffusiveorviscousscaling.
2.1. Hyperbolic versus parabolic scaling 77
Equivalence of alternative formulations
So far, we have considered the transformation of the evolution equations7, with-out losing a word abwith-out accompanying initial and boundary conditions. However, evolution equations alone are mathematically not reasonable and must be comple-mented by further conditions to obtain well-posed problems with unique solutions.
Therefore we now give a complete statement of the associated IBVPs8.
In particular we deal with the question whether the solution of the lattice-Boltzmann system can be reconstructed from the solution of the scalar equation (2.21). We establish the equivalence of the subsequent IBVPs in the sense, that the solution of one IBVP induces solutions to the other ones. For each IBVP we present two different boundary conditions, that are assumed to be homogeneous for simplic-ity. The IBVPs are considered over a time-space domain XT := [0, T]×[0, L] with
IBVP for the equivalent moment system:
EQ : ∂tu+ ∂xφ = q
It is clear that (2.22) and (2.23) must be equivalent, since the variables (f1,f2) and (u, φ) are related to each other by an invertible linear transformation. In
7Reduction of the D1P2 lattice-Boltzmann system into a scalar equation via the equivalent moment system.
8Shortcut for initial boundary value problem. Further abbreviations: EQ = (evolution) equa-tion, IC = initial condition and BC = boundary condition.
particular, the bounce-back-type boundary condition with sign flipping yields at the left boundary
f2(t,0) =−f1(t,0) ⇔ f1(t,0) +f1(t,0) = 0 ⇔ u(t,0) = 0,
what corresponds to homogeneous Dirichlet boundary condition for the mass mo-mentu. Similarly, the second boundary condition results in homogeneous Dirichlet boundary conditions forφ:
f2(t,0) =f1(t,0) ⇔ f1(t,0)−f2(t,0) = 0 ⇔ ǫφ(t,0) = 0 ⇔ φ(t,0) = 0. The equivalence of (2.23) and (2.24) turns out to be more delicate, as one is obliged to differentiate (loss of information), when deriving the scalar equation from the equivalent moment system. So, the reverse way, from (2.24) to (2.23), must comprise an integration giving rise to an undetermined integration constant. Before we settle this problem in the next proposition, let us check how the initial and boundary conditions of (2.23) translate into those of (2.24).
From the first evolution equation of (2.23) we obtain the following relation between the initial data p and h:
h(x) =∂tu(0, x) =q(0, x)−∂xφ(0, x) =q(0, x)−∂xp(x).
As far as the BC ii) in (2.23) is concerned, φ(t,0) = 0 implies∂tφ(t,0) = 0 and this gives BC ii) in (2.24) using equation (2.19). In the general case of inhomogeneous boundary conditions, i.e. φ(t,0) =r(t), we get
f u(t,0)
−τ ∂xu(t,0) =r(t) +ǫ2τr(t)˙ .
Proposition 2.1. Suppose that u∈ C2(XT) is a solution of the IBVP (2.24) satis-fying either boundary condition i) or ii). Assume moreover that the compatibility
condition Z L
0
h
q(0, ξ)−h(ξ)i
dξ = 0 (2.25)
is fulfilled in the case of ii). Then there exists a function φ ∈ C1(XT), such that (u, φ) becomes a solution of IBVP (2.23) satisfying boundary condition i) or ii) respectively. In the case of boundary condition i) φis uniquely determined up to a purely time-dependent term.
Proof: Let us defineφin such a way that the second evolution equation in (2.23) is automatically satisfied. A slight reorganization of this equation yields
∂tφ+ǫ−2ωφ=ǫ−2ωf(u)−ǫ−1∂xu . (2.26) The solution of this ODE forφwith respect totis given by thevariation of constants formula
φ(t, x) = (
C+ Z x
0
h
q(0, ξ)−h(ξ)i dξ
) e−ωt/ǫ2 +ǫ−2ωe−ωt/ǫ2
Z t
0
eωθ/ǫ2h
f u(θ, x)
−τ ∂xu(θ, x)i
dθ (2.27)
2.1. Hyperbolic versus parabolic scaling 79
with a constantCto be specified later. Now we have to check that the first equation of (2.23) holds true with φ defined as above. By definition (2.27) we see that φ∈ C1(XT) if u ∈ C2(XT). Furthermore, it can be read off from the formula that the mixed derivative ∂t∂xφ exists and is continuous. Therefore we are allowed to differentiate (2.26) with respect tox:
ǫ2τ ∂t(∂xφ) +∂xφ=f′(u)∂xu−τ ∂x2u .
We have multiplied the differentiated equation byǫ2τ in order to compare it more easily with the slightly rearranged evolution equation of (2.24):
ǫ2τ ∂t(q−∂tu
| {z }) +q−∂tu
| {z }=f′(u)∂xu−τ ∂x2u .
Obviously, the quantity q−∂tu satisfies the same ODE with respect to t as ∂xφ.
Moreover we infer from (2.28) that∂xφ(0, x) =q(0, x)−h(x) =q(0, x)−∂tu(0, x).
Due to the agreement of the initial conditions we can conclude from theuniqueness and existence theorem of Picard-Lindel¨of9 that
∂xφ(t, x) =q(t, x)−∂tu(t, x) for allt≥0. But this equation is equivalent to the first evolution equation of (2.23).
Let us now check the boundary conditions: In the case of BC i),φis not subject to boundary conditions. Thus the integration constant C is freely selectable. Being multiplied by the factor e−ωt/ǫ2, it is responsible for the time dependent term.
The case of BC ii) – corresponding to homogeneous natural boundary conditions for u – results in homogeneous Dirichlet conditions for φ, if C = 0 and if the compatibility condition (2.25) is fulfilled:
9The important theorem of Picard-Lindel¨of statesgrosso modo, that the initial value problem z(0) =z0, z(t) =˙ F`
t, z(t)´
has aunique solution in a neighborhood of t= 0, if the function F is continuous w.r.t. its first and Lipschitz-continuous w.r.t. its second argument. Furthermore the theorem gives an lower estimate for the interval of existence in dependence of the Lipschitz constant. But this is here of no importance, since the equation is linear and the explicit formula (2.27) provides the solution for allt∈[0, T]. Therefore we only need the uniqueness statement for the above reasoning.
Remark 2.1. (Compatibility condition) Compatibility conditions for the pre-scribed data (sources, initial and boundary values) often arise to avoid possible conflicts with the evolution equation or other constraints.
For IBVP (2.24) the source q and the initial values for u and ∂tu are prescribed.
From its solutionuand the prescribed data we have constructed a functionφsubject to homogeneous Dirichlet boundary conditions, such that (u, φ) solves IBVP (2.23).
Now, the first evolution equation of (2.23) connects the quantities ∂xφ, ∂tu and q.
Integrating this equation over the full space interval [0, L] yields Z L
0
∂ξφ(t, ξ) dξ =φ(t, L)−φ(t,0) = Z L
0
[q(t, ξ)−∂tu(t, ξ)]dξ .
For t= 0 we obtain that the difference of the initial boundary values ofφ must be equal to RL
0 [q(0, ξ)−h(ξ)]dξ. So we retrieve compatibility condition (2.25) in the special case of φ(t,0) =φ(t, L) = 0.
What is the physical meaning of this compatibility condition? To answer this, we take a glance on the target equation (2.16), that permits a clear physical interpre-tation in contrast to (2.23) and (2.24) if we identify u – for instance – with a mass density. By introducing the physical flux ϕ := f(v)−ν∂xv, we can derive in the same manner as above the following equation
d dt
Z L
0
v(t, ξ) dξ ≡ Z L
0
∂tv(t, ξ) dξ = ϕ(t, L)−ϕ(t,0) + Z L
0
q(t, ξ) dξ . The left hand side is the time derivative of the total mass, i.e. the instantaneous change in total mass. Intuitively, it must be equal to what is flowing out or coming in at the boundary points 0, L (represented by ϕ(t, L)−ϕ(t,0)) plus the mass production rate represented by the space integral over the source function q. Thus the condition expresses conservation of mass.
Since uand φapproximatevand ϕrespectively, the existence of a similar compat-ibility condition involving u, φ is not surprising. Nevertheless, it is remarkable to hit on the same compatibility condition althoughϕis differently defined fromφvia the ODE (2.19). Finally note, that (2.24) alone with BC ii) does not enforce the compatibility condition10.
Remark 2.2. (Gauge freedom)The fact thatφcan not be uniquely defined from (2.24) accompanied by BC i) reflects a property of the lattice-Boltzmann system (2.22) including BC i), that one might interpret as a sort of gauge freedom for the quite artificial population functions. If (f1,f2) is a solution of the IBVP (2.22) subject to BC i), then (˜f1,˜f2) with ˜f1 :=f1−ce−ωt/ǫ2 and ˜f2 :=f2+ce−ωt/ǫ2 solves (2.22) with BC i) as well but satisfies different initial conditions.
However, from the macroscopic perspective the crucial point is, that (f1,f2) and (˜f1,˜f2) have the same mass momentu. Hence they approximate the samev, which is of physical relevance as solution of the target equation. Moreover, the first moments φ= 1ǫ(f2−f1) and ˜φ= 1ǫ(˜f2−˜f1) =φ+ 2cǫ−1e−ωt/ǫ2 converge also for ǫ→0 to the same function albeit being different.
10Instead we find: RL
0 [q(t, x)−∂tu(t, ξ)]dξ = −ǫ2τRL
0 [∂tq(t, x)−∂t2u(t, ξ)]dξ = O(ǫ2). This relation does not entail a condition for the prescribed data, since∂t2uis not initialized.