Boltzmann Methods
1.2 Translational invariance and dimensional reduction
It is important to note that the singular parameter ǫ occurring in the lattice-Boltzmann equation (e.g. in front of the relaxation term) has been completely re-moved in the discretization above. The reason for this is, that the time step ∆t is coupled to ǫin a specific way which melts two a priori independent limit processes into a single, simultaneous limit process. Therefore (1.34) represents no consistent discretization of (1.32) but aims directly towards an approximate solution of the target equation.
1.2 Translational invariance and dimensional reduction
It has already been mentioned in the preceding section that 2D and 1D lattice-Boltzmann velocity models are related to each other. In this section we exem-plarily show how D1P3 algorithms are extracted from D2P9 algorithms. Gener-ally, two-dimensional Boltzmann schemes result into one-dimensional lattice-Boltzmann schemes if certain symmetry conditions are fulfilled. Therefore the lat-ter ones provide appropriate starting points for theoretical investigations of the first ones – not just due to structural similarities but, what weighs more, as a kind of simplifying reformulation.
The first paragraph discusses shear flows as translationally invariant solutions of the incompressible Navier-Stokes equation (INSE). For this class of flows it is possible to boil down the nonlinear, two-dimensional INSE to become the linear diffusion equation in one space dimension. The following paragraphs are devoted to the D2P9 model where we try to mimick this process of simplification. Besides the reasons mentioned above it is interesting to check, whether the D2P9 lattice-Boltzmann algorithm exhibits similar symmetries as those equations, that can be approximated by it (Stokes, Oseen, Navier-Stokes).
Definition 1.1. (Translational invariance of sets and functions)
i) A subset Ω ⊂ Rn is called translationally invariant with respect to the dis-placement vector d∈Rn\ {0} iff: ∀α∈R: x∈Ω ⇒ x+αd ∈Ω.
ii) Let Ω⊂Rn betranslationally invariant with respect tod. A functionf : Ω→ Rn is called translationally invariantor simply constant in the direction of d iff: ∀α∈R: f(x) =f(x+αd).
Obviously, translationally invariant subsets correspond to stripes parallel to d.
These sets are unbounded and do not permit a finite, regular discretization with a constant grid spacing. This however would be very desirable with regard to com-puter simulations. Therefore we introduce two planar manifolds. In the following, the constants H, L denote some positive numbers.
planar torus: T :=R/L×R/H planar cylinder: Z := [0, L]×R/H The torus corresponds to the rectangle [0, L]×[0, H] where opposite sides are identi-fied. This manifold is translationally invariant in any direction. For flow simulations it is realized by a rectangle with periodic boundary conditions inx- andy-direction.
If only one pair of opposite sides is identified, the planar cylinder is obtained. Here
the sides parallel to thex-axis are chosen such that the cylinder becomes translation-ally invariant in y-direction. The cylinder models a pipe where periodic boundary conditions are assumed at the inlet and outlet to avoid more complicated inflow and outflow conditions.
Shear Flows
Shear flows are distinguished by the fact that the streaming velocity is everywhere parallel to a fixed flow direction. Moreover the incompressibility enforces the veloc-ity field to be translationally invariant48 in this direction. It turns out that shear flows can not only be described as solutions of the incompressible Navier-Stokes equation (INSE) but also as solutions of the simpler diffusion equation. This cir-cumstance facilitates the finding of analytic solutions. Two prominent examples are the Poiseuille flow and the Couette flow. Both are frequently used as simple benchmarks for numeric Navier-Stokes solvers.
Definition 1.2. A solution of the INSE is called ashear flow iny-direction, if the x-component u of the velocity v= uv
is identically zero.
The incompressiblity condition∇ ·v=∂xu+∂yv= 0 implies immediately that∂yv must vanish, becauseu anda fortiori∂xuis zero by assumption. Hence,v must be independent of y and the velocity vectorv simplifies in the following way:
v(t, x, y) =v(t, x) = 0
v(t, x)
(1.35) Proposition 1.3. The incompressibility condition entails that any solution of the INSE being a shear-flow in y-direction is constant in y-direction.
We conclude that the velocity field of a shear flow is completely described by a single scalar function of one spatial coordinate. This justifies the commonly used termquasi-1d flow. v is often referred to as the velocity profileorflow profile.
Plugging (1.35) into the INSE leads to the evolution equation for v:
∂t
Noting that the nonlinear term disappears, one obtains 0 =−∂xp ∧ ∂tv − ν ∂x2v = −∂yp.
The independence of the velocity with respect to the y-coordinate carries over to the pressure p. This entails ∇p = 0 obliging the pressure to be constant. Being uniquely determined up to an additive constant, the pressure can be set to 0.
Evidently, the flow profilev appears as a solution of theone-dimensional diffusion equation. The result is summarized more precisely by the following proposition.
48In principle, this requires the flow domain to be translationally invariant as well.
1.2. Translational invariance and dimensional reduction 45
Proposition 1.4. (Conservation of translational invariance) i) Consider the IVP for the INSE in the torus T:
∂tv+v· ∇v−ν∆v=−∇p+g,
∇ ·v= 0, v(0,·) =v0.
If v0(x, y) = 0
v0(x)
and g(t, x, y) = 0
g(t, x)
, (∗) there exists a global solution that is constant in y-direction, provided the dif-fusion problem
∂tv−ν∂x2v=g v(0,·) =v0 admits a global solution.
ii) Consider the IBVP problem for the INSE in the cylinder Z:
∂tv+v· ∇v−ν∆v=−∇p+g
∇ ·v= 0 v(0,·) =v0
v(t, x,0) =w0(t, x) v(t, x, H) =wH(t, x) If (∗) is satisfied and moreover for some a∈R
w0(t, x) = a
w0(t)
, wH(t, x) = a
wH(t)
,
there exists a global solution that is constant in y-direction provided the ad-vection-diffusion problem
∂tv+a∂yv−ν∂y2v=g v(0,·) =v0
v(t,0) =w0(t) v(t, H) =wH(t) has a global solution.
Proof: The solution of the Navier-Stokes problem is obtained by settingv(t, x, y) = v(t, x) = v(t,x)0
in first case andv(t, x, y) =v(t, x) = v(t,x)a
in the second one.
Remark 1.1. If the given data, i.e. the initial velocity field and the exterior force, are constant iny-direction, the solution for any IVP of the INSE on a translationally invariant domain inherits this property. Therefore the INSE preserves translational invariance.
Remark 1.2. Parallel shear flows are solutions of the Stokes and the Navier-Stokes equation, since the nonlinear, convective term of the Navier-Stokes equation van-ishes in the case of the special velocity field.
Reduction of the D2P9 lattice-Boltzmann algorithm
Let us consider the lattice-Boltzmann scheme with the D2P9 velocity model on a rectangular grid discretizing the torus. I, J ∈Ndenote the number of grid nodes in x- and y-direction respectively. The populations Fk withk∈ {1, ...,9} are indexed as indicated in table 1.1. Moreover the Fk’s are taken as functions of the time step indexn and two spatial indicesi, j specifying the grid node the Fk’s belong to:
Fk:N0×Z/IZ×Z/JZ→R.
The equilibrium E is assumed to depend on the mass moment R (interpreted as pseudo-density) and the moments U, V of first order (interpreted as velocity com-ponents) like (1.30) and (1.31). For convenience we set
Ek(n, i, j) :=Ek R(n, i, j), U(n, i, j), V(n, i, j)
with R:=P
kFk, U :=P
kFkskx, V :=P
kFksky. )
(1.36) The populations evolve in time according to equation (1.34) that we write in the subsequent form:
Fk(n+ 1, i, j) = (1−ω)Fk(n, i−skx, j−sky) + ωEk(n, i−skx, j−sky)
⇔ Fk(n+ 1, i+sxk, j+sky) = (1−ω)Fk(n, i, j) + ωEk(n, i, j)
) (1.37) The choice of the spatial index sets Z/IZ, Z/JZ encodes directly the periodic boundary conditions along all sides of the rectangular domain.
The following property of the lattice-Boltzmann algorithm is as simple as funda-mental.
Proposition 1.5. (Conservation of translational invariance)
If the populations are initialized such that they do not vary withj, then they remain independent of j for all iterations (translational invariance in y-direction).
Proof: The proof is done by recurrence. For n = 0 the assertion is true by as-sumption. Let us now conclude that it remains true from the nth to the (n+ 1)th iteration:
By hypothesis of induction the Fk(n,·,·)’s on the right hand side are constant with respect to j. Thus the Ek(n,·,·)’s do not vary with j either, since the equilibrium is assumed to depend not explicitly onj. So it depends onj only via the moments R, U, V being however linear combinations of theFk(n,·,·)’s. Hence the right hand side is independent ofjand so must be the left hand side. This is equivalent to the claim that the populations of the (n+ 1)th iteration are independent ofj.
Remark 1.3. Since the populationsFkare allowed to attain arbitrary real values, the set of all admissible states pertaining to the lattice-Boltzmann scheme forms a vector space being isomorphic to R9IJ. Evidently, the subset of states which are independent of j represents a subspace of dimension 9I. The proposition is equivalent to the statement, that this subspace is invariant under the evolution operator of the lattice-Boltzmann scheme.
1.2. Translational invariance and dimensional reduction 47
Let us suppose that the assumption of proposition 1.5 is fulfilled. Consequently, the index j can be omitted in (1.37). Illustratively spoken, this means that the rect-angular two-dimensional grid is replaced by a one-dimensional grid corresponding to a cross-section along the x-axis. The accompanying one-dimensional evolution equationsbecome:
F1,2,8(n+ 1, i) = (1−ω)F1,2,8(n, i−1) + ω E1,2,8(n, i−1), F3,7,9(n+ 1, i) = (1−ω)F3,7,9(n, i) + ωE3,7,9(n, i), F4,5,6(n+ 1, i) = (1−ω)F4,5,6(n, i+ 1) + ω E4,5,6(n, i+ 1).
(1.38)
The nine populations are grouped into three triples:
the right-moving populations F1,F2,F8 withskx= 1, the resting populations F3,F7,F9 withskx= 0,
the left-travelling populationsF4,F5,F6 withskx=−1.
Obviously, the populations of each triple share the same evolution equation. There-fore one might be tempted to reduce the number of populations and equations by taking simply one equation out of every triple. However, this idea does not work out, since the evaluation of the equilibrium requires the computation of the mo-ments R, U and V. These, however, can not be computed by the knowledge of three populations alone and therefore the system is not closed.
R = F3+F7+F9 + F1+F2+F8 + F4+F5+F6 U = F1+F2+F8 − F4+F5+F6
V = F2−F8 + F3−F7 + F4−F6
Actually, the full knowledge of the 9 populations is not necessary. The moments R, U and V are available as soon as the following quantities are known:
A+ :=F1+F2+F8 D+:=F2−F8 A0 :=F3+F7+F9 D0:=F3−F7
A− :=F4+F5+F6 D−:=F4−F6
⇒
R = A+ + A0 + A−
U = A+ − A−
V = D+ + D0 + D−
(1.39) Each of these 6 linear combinations is only composed of those populations that belong to the same triple. In order to obtain the evolution system for the new quantities, the equations (1.38) have to be combined in the same manner. Since R, U, andV can be expressed only in terms of the new quantities, the system is closed. Therefore we are able to formulate the next proposition.
Proposition 1.6. Consider the D2P9 lattice-Boltzmann scheme on a rectangular grid discretizing the torus. If the initial conditions are constant iny-direction, then the lattice-Boltzmann algorithm can be reduced to a one-dimensional scheme using 6 instead of 9 populations without affecting the numeric values of the three moments R,U and V.
The elimination of further populations is only possible under extra assumptions.
We distinguish between two possibilities:
i) (pure) shear mode: R= 0 ∧ U = 0 ii) (pure) acoustic mode: V = 0
Proposition 1.7. Consider the D2P9 lattice-Boltzmann scheme on a rectangular grid discretizing the torus.
i) Shear mode: Suppose that the initial conditions are constant in y-direction and that moreover the moments R and U vanish for all iterations. Then the D2P9 scheme can be reduced to a D1P3 scheme without affecting the numeric value of V.
ii) Acoustic mode: Suppose that the initial conditions are constant iny-direction and that moreover the moment V vanishes for all iterations. Then the D2P9 scheme can be reduced to a D1P3 scheme without affecting the numeric values of R, U.
Proof:
ad i) Using the assumption R= 0, U = 0 and setting
E+D(V) := E2(0,0, V)− E8(0,0, V), E0D(V) := E3(0,0, V)− E7(0,0, V), E−D(V) := E4(0,0, V)− E6(0,0, V), the following system is directly derived from (1.38):
D+(n+ 1, i) = (1−ω)D+(n, i−1) + ωE+D V(n, i−1) , D0(n+ 1, i) = (1−ω)D0(n, i) + ωE0D V(n, i)
, D−(n+ 1, i) = (1−ω)D−(n, i+ 1) + ωE−D V(n, i+ 1)
.
SinceV =D++D0+D−, the system is closed and corresponds to the evolution equation of the D1P3 algorithm.
ad ii) Proceeding analogously as above
E+A(R, U) = E1(R, U,0) + E2(R, U,0) + E8(R, U,0), E0A(R, U) = E3(R, U,0) + E7(R, U,0) + E9(R, U,0), E−A(R, U) = E4(R, U,0) + E5(R, U,0) + E6(R, U,0),
we stumble again on the evolution equation of the D1P3 lattice-Boltzmann scheme:
A+(n+ 1, i) := (1−ω)A+(n, i−1) + ωE+A R(n, i−1), U(n, i−1) , A0(n+ 1, i) := (1−ω)A0(n, i) + ωE0A R(n, i), U(n, i)
, A−(n+ 1, i) := (1−ω)A−(n, i+ 1) + ωE−A R(n, i+ 1), U(n, i+ 1)
. Due to R=A++A0+A− andU =A+−A− the system is closed.
In the sequel we have to verify that the assumption of the proposition can be fulfilled. So it must be shown that the D2P9 algorithm can be initialized so thatR and U orV remain zero over all iterations.
1.2. Translational invariance and dimensional reduction 49
Remark 1.4. (Boundary conditions)In order to study the translation of bound-ary conditions from the full D2P9 scheme to the reduced D1P3 scheme, we consider a rectangular grid discretizing the cylinder. The populations are then functions of the following type:
Fk :N0× {1, ..., I} ×Z/JZ→R
{1, ..., I}indicates that inx-direction no periodic boundary conditions are applied.
Hence the populations F1,F2,F8 at the left boundary nodes (i = 1) and the pop-ulations F4,F5,F6 at the right boundary nodes (i = I) cannot be updated by the standard propagation (1.37). A workaround is offered by the bounce-back rule, which approximates homogeneous Dirichlet (no-slip) boundary conditions for the target equation along the walls (x= 0 andx=L) of the cylinder.
In this case the populationsF1,F2,F8 and F4,F5,F6 are updated by retroflection49 of the collision products, that would otherwise leave the domain. More precisely:
Fk(n+ 1, i, j) = (1−ω)F¯k(n, i, j) +ωE¯k(n, i, j). (1.40) Here ¯kdenotes the index of the population whose velocity vector is the negative of the velocity vector associated with the populationk, i.e. sskx¯¯
ky
=− sskxky
. Note that the retroflection involves collision products and populations pertaining to the same node. Therefore the translational invariance is not affected and proposition 1.5 is also valid in the case of the cylinder.
Similarly, the one-dimensional equations (1.38) are supplemented by
Fk(n+ 1, i) = (1−ω)F¯k(n, i) + ωE¯k(n, i), (1.41) which has to be applied whereever (1.38) fails due to the presence of boundaries.
Under the assumptions of proposition 1.7 let us now derive the boundary conditions for the D1P3 scheme from the above bounce-back rule for the D1P2 scheme. Writing down the bounce-back rule forF2,F8
F2(n+ 1, i) = (1−ω)F6(n, i) + ωE6(0,0, V(n, i)) F8(n+ 1, i) = (1−ω)F4(n, i) + ωE4(0,0, V(n, i)) we find
F2(n+ 1, i)−F8(n+ 1, i) = (1−ω)h
F6(n, i)−F4(n, i)i + ωh
E6(0,0, V(n, i))− E4(0,0, V(n, i))i
⇒ D+(n+ 1, i) =−h
(1−ω)D−(n, i) + ω E−D V(n, i)i and similarly D−(n+ 1, i) = −h
(1−ω)D+(n, i) + ωE+D V(n, i)i .
As expected, we obtain forD+,D−bounce-back type rules differing from the original bounce-back rule only by a flipping of sign. Proceeding analogously, the same boundary conditions are also found forA− and A+.
49In opposition to thereflection, theretroflectionflips the whole velocity vector and not only its normal component. So, a particle hitting a wall with the velocityvin, is retroflected if it leaves the wall withvout=−vin, whereas it is reflected ifvout=vin−2n·vin. Herendenotes the unit normal vector at the position of impact, that points into the wall.
Remark 1.5. Clearly, all statements remain true mutatis mutandis, if we assume translational invariance in x-direction. Moreover, if the lattice-Boltzmann algo-rithm is driven by some external source then the translational invariance of the initial state is only conserved if the force is also translationally invariant with re-spect to the same direction.
The Stokes equilibrium
In order to become more concrete concerning the reduction of the D2P9 algorithm we have to specify the equilibrium. Let us start with the Stokesequilibrium being a linear function of the moments R, U and V introduced in (1.36).
EkS(R, U, V) :=wk R + 3U sxk + 3V syk
. (1.42)
Observe that this equilibrium corresponds to the linear part of (1.31). If (1.42) is combined with the parabolic scaling,U and V converge to thex- andy-component of the velocity while the pseudodensity R approximates the pressure.
Lemma 1.1. (Shear mode) Consider the D2P9 scheme with the Stokes equilib-rium on an one-dimensional grid with either periodic or bounce-back boundary con-ditions. Assume that the following relations hold true at the initialization n = 0.
Then they hold true for all iterations:
i) F1(n, i) + F2(n, i) + F8(n, i) = 0 ii) F3(n, i) + F7(n, i) + F9(n, i) = 0 iii) F4(n, i) + F5(n, i) + F6(n, i) = 0
Proof: The verification is done by recurrence. According to the assumption the relations are true forn= 0. So, their correctness has to be checked for the (n+ 1)th step provided their validity at thenth step is given.
From i) - iii) we obtain immediately
R(n, i) = 0 and U(n, i) = 0.
Therefore we conclude:
E1(n, i) +E2(n, i) +E8(n, i)
=E1S 0,0, V(n, i)
+E2S 0,0, V(n, i)
+E8S 0,0, V(n, i)
= 0 + 121V(n, i) − 121V(n, i) = 0. Since this relation together with i) holds for all spatial indices i, we find by using the evolution equation (1.38):
F1(n+ 1, i) + F2(n+ 1, i) + F8(n+ 1, i) = (1−ω)
F1(n+ 1, i−1) + F2(n+ 1, i−1) + F8(n+ 1, i−1)
+ ω
E1(n+ 1, i−1) + E2(n+ 1, i−1) + E8(n+ 1, i−1)
= (1−ω)·0 +ω·0 = 0.
1.2. Translational invariance and dimensional reduction 51
The crucial point is, that the collision products updating the populations occurring in i) come from the same neighbor node. Therefore the recurrence assumption can be applied. Similarly ii) and iii) are proved. In the case of retroflection, equation
(1.41) has to be used instead of equation (1.38).
Letv0 =v0(x) be an initial profile of a shear flow in y-direction and let the coordi-nates of a regular one-dimensional grid be given by{xi}i∈Z/IZ and{xi}i∈{1,...,I} for the torus and for the cylinder respectively.
Proposition 1.8. The following initializations of the D2P9 scheme satisfy the assumption of the preceding lemma.
i) Fk(0, i, j) = Fk(0, i) = EkS 0,0, v0(xi)
(1.43) ii) Fk(0, i, j) = Fk(0, i) = EkS 0,0, v0(xi)
+ γ ∂xv0(xi)wkskxsky (1.44) Remark 1.6. The additional correction term in (1.44) provokes a smoother initial behavior. γ is a coefficient being of no importance in this context.
Now, we are just in the situation, where proposition 1.7 i) is applicable. Utilizing the same notation as in its proof, the Stokes equilibrium reduces to
E+DS(V) = 16V , E0DS(V) = 23V , E−DS(V) = 16V . (1.45) It should be noticed that we use the D1P3 velocity set{−1, 0, +1} ≡ {−, 0, +}in order to index quantities referring to the D1P3 velocities. The initializations (1.43) and (1.44) take the following form for the D1P3 scheme:
i) Dσ(0, i) = EσDS v0(xi)
(1.46) ii) Dσ(0, i) = EσDS v0(xi)
+ 181 γ σ ∂xv0(xi) (1.47) D0 is interpreted as the rest population; D+,D− represent populations moving to the right and left respectively. Their mass moment (sum of D−,D0 and D+) is ex-actly V, which is the first order moment of the D2P9 scheme approximating the velocity component iny-direction.
As in the D2P9 case the rest population is associated with the strongest weight w0 = 23, the other two populations are equally weighted, i.e. w+ =w− = 16, such that the isotropy of the scheme is not destroyed andP
σwσ = 1. The next theorem recapitulates the whole result we have proved so far.
Theorem 1.1. (Reduction theorem for shear modes) Consider the D2P9 scheme with the Stokes equilibrium (1.42) on a rectangular grid discretizing the torus or the cylinder. Furthermore consider the D1P3 scheme with the equilibrium defined in (1.45) on the one-dimensional grid representing the cross-section of the rectangular grid in x-direction. Suppose the D2P9 scheme is initialized by (1.43) or (1.44) and the D1P3 scheme is initialized by (1.46) or (1.47) respectively. Then both schemes produce identical numeric results for the approximate flow profile V. Remark 1.7. (Forcing) In the case of external forces, lemma 1.1 remains only true if the source term, which has to be added to the right hand sides of the
evolution equation (1.38) and the bounce-back rule (1.41), is of the formGy(n, i)sky. Physically, this reflects the fact that a shear flow iny-direction can only be driven by a force which points in this direction and varies spatially only in the perpendicular direction.
Remark 1.8. If we consider the D2P9 scheme on an one-dimensional grid with I nodes, then the global state space is a vector space of dimension 9I. The equations i), ii) and iii) of lemma 1.1 define a 6I dimensional subspace, which is asserted to be invariant under the evolution operator. Note, that this subspace is contained in the larger subspace of dimension 7I defined by R = 0 and U = 0. However, this latter subspace seems to be not invariant under the evolution. An explanation for this can be obtained by the asymptotic analysis yielding an interpretation for quantities like F1+F2+F8. It turns out that they represent components of the stress tensor, which have to vanish in order to be consistent with the shear flow and and to keep R and U equal to zero.
Remark 1.9. It is possible to prove a stronger version of lemma 1.1 claiming additionally, that F1,F5 and F9 do not change if they are initialized by zero. Note furthermore, that it is not possible to reconstruct the populations of the D2P9 scheme in an unique way from the those of the D1P3 scheme. In that sense the schemes are not equivalent.
Let us now turn to the acoustic modes:
Lemma 1.2. (Acoustic mode) Consider the D2P9 scheme with the Stokes equi-librium on an one-dimensional grid with either periodic or bounce-back boundary conditions. Assume that the following relations hold true at the initialization n= 0.
Then they hold true for all iterations:
i) F2(n, i) − F8(n, i) = 0 ii) F3(n, i) − F7(n, i) = 0 iii) F4(n, i) − F6(n, i) = 0
Proof: The verification is analogous to the proof of lemma 1.1.
i)-iii) yields immediately
V = (F2−F8) + (F3−F7) + (F4−F6) = 0 + 0 + 0 = 0 From this we infer:
E2(n, i)−E8(n, i) = E2S R(n, i), U(n, i),0
− E8S R(n, i), U(n, i),0
= 361 R(n, i) + 3U(n, i) + 0
− 361 R(n, i) + 3U(n, i)−0
= 0
In the same way it is shown that: E3(n, i)−E7(n, i) = 0 E4(n, i)−E6(n, i) = 0.
Since this relation together with i) holds for all spatial indices i, we find by using the evolution equation (1.38):
F2(n+ 1, i)−F8(n+ 1, i) = (1−ω)
F2(n, i−1)−F8(n, i−1) + ω
E2(n, i−1)−E8(n, i−1)
= (1−ω)·0 +ω·0 = 0
1.2. Translational invariance and dimensional reduction 53
The configuration of the cylinder implies thatF2 is updated by retroflection iffF8 is updated by retroflection. Therefore we obtain in this case at the left boundary (i= 1):
F2(n+ 1,1)−F8(n+ 1,1) = (1−ω)
F¯2(n,1)−F¯8(n,1) + ω
E¯2(n,1)−E¯8(n,1)
= (1−ω)
F6(n,2)−F4(n,2) + ω
E6(n,2)−E4(n,2)
= (1−ω)·0 +ω·0 = 0
Similarly, ii) and iii) are shown.
Letρ0 =ρ0(x) be the initial distribution of the pseudodensity andu0(x, y) = u00(x) the initial velocity. If the D2P9 algorithm is initialized by
Fk(0, i, j) = Fk(0, i) = EkS ρ0(xi), u0(xi),0
, (1.48)
then the hypothesis of the preceding lemma is fulfilled and proposition 1.7 ii) can be employed. In the case of the acoustic modes the Stokes equilibrium reduces to
EσAS(R, U) :=wσ R+ 3σ U
, σ ∈ {−,0,+}. (1.49) Consequently, the D1P3 scheme is initialized by
Aσ(0, i) = EσAS ρ0(xi), u0(xi),0
. (1.50)
The following theorem is the acoustic mode version of the main result, theorem 1.1.
Theorem 1.2. (Reduction theorem for acoustic modes)Consider the D2P9 scheme with the Stokes equilibrium (1.42) on a rectangular grid discretizing the torus or the cylinder. Furthermore consider the D1P3 scheme with the equilibrium defined by (1.49) on the one-dimensional grid corresponding to the cross-section of
Theorem 1.2. (Reduction theorem for acoustic modes)Consider the D2P9 scheme with the Stokes equilibrium (1.42) on a rectangular grid discretizing the torus or the cylinder. Furthermore consider the D1P3 scheme with the equilibrium defined by (1.49) on the one-dimensional grid corresponding to the cross-section of