Consistency of the truncated expansion

for allh < h₀ andn∈N_h

ke(0)k²a,Ωh+

n−1

X

k=0

(2 + 1

h²)kη(k) +Υ(k)k^2V_h≤h²1

Te^−C0T. (4.63) which allows us to apply Lemma 2 to the inequality (4.58). Hence

ke(n)ka,Ωh≤C

ke(0)ka,Ωh+h^κ3−2+h^κ1−2

≤C

h^κ2+h^κ3−2+h^κ1−2 (4.64) and the result follows.

Again, we apply the general result of the theorem above to our specific predic-tion funcpredic-tion ˜f,

Theorem 19 (Convergence of f˜ on bounded domains)

Let f be the solution of a given lattice Boltzmann algorithm E³_h which fulfills the assumptions of Theorem 15. Assume the prediction function f˜ satisfies Theorem 9 withκb>5/2, Then

kf˜−fk^Vh =O(h^σ˜), σ˜ = min(κ−2, κ₀, κ_b −3

2) =κ_b−3 2,

and there existh-independent constantsC >0andh₀ >0so that for allh < h₀ andn∈N_h,

kp(tn)−P(n)kΩh ≤Ch^{min(2,κ0−2,κb−72)}, ku(tn)−U(n)kΩh≤Ch^{min(2,κ0−1,κb−52)}

for the moments (4.21) of the lattice Boltzmann solution, where p(tn),u(tn)∈ Gh are the restrictions of the solution p,u of (2.1), (2.3) to the grid.

Proof: Since [κ, κ₀, κb+ 1/2]^T ∈ord_E³( ˜f), applying Theorem 18 to ˜f finally leads to this theorem by takingκ≥6,κ₀ ≥2 and 1< κb−³₂ < ³₂ into account..

4.5 Consistency of the truncated expansion

To assess the consistency order of ˜f to the lattice Boltzmann solution, we have to find the undetermined orders of ˜f inA, which are related to the undetermined orders of ˜f in A0. Recall that the slight difference of A0 and A lies on the additional condition ku(s)k∞,Ωh = O(h). Therefore there is some relation between Ms and M⁰s, which is described in the following two Lemmas and holds for ˜f.

Lemma 8 Let E^m_h be a lattice Boltzmann algorithm E^m_h and s ∈ A0. If k ∈ Ms(E^m_h) and k≥1, then also k∈ M⁰s(E^m_h).

Proof: Since there is a γ ∈ A satisfying 0 ≤ c0 ≤ kγk^Vh ≤ c1 such that s+h^kγ∈ Aand

kE⁽ⁱ⁾_h (s+h^kγ)k^Vh =O(kE⁽ⁱ⁾_h (s)k^Vh), 1≤i≤m.

this lemma follows if s+h^kγ ∈ A0 is proved.

Since s ∈ A0, there is constant C^′ such that ku(s)k∞,Ωh ≤ C^′h. Using the triangle rule and ku(γ)k^∞_Ωh≤ kγk^Vh, we find that

ku(s+h^kγ)k∞,Ωh≤ ku(s)k∞,Ωh+h^kku(γ)k∞,Ωh ≤C^′h+h^kc₁ ≤max(C^′, c₁)h.

In some particular cases, both sets are even equivalent.

Proposition 2 For a given lattice Boltzmann algorithm and some s∈ A0, we have M⁰s =Ms if infMs≥1.

Proof: This can be easily proved with the help of Lemma 8.

Since a direct checking of the element of Mf˜ is very difficult, we adopt a different strategy. To obtain an upper bound of the consistency order, we explicitly verify that a certain order is undetermined giving rise to a specific element of Mf˜. To achieve a lower bound, we use the estimated convergence order in connection with Theorem 5.

We introduce three numbers for the regular expansion ˜f. κ^∞= sup

κ∈ordEm( ˜f)

κ₁, κ^∞₀ = sup

κ∈ordEm( ˜f)

κ₂, κ^∞_b = sup

κ∈ordEm( ˜f)

κ₃. In case that m= 2,κ^∞_b is dropped out.

4.5.1 Periodic cases

Proposition 3 (Mf˜ for nonlinear periodic cases)

Provided that the conditions for the lattice Boltzmann scheme E²_h and the pre-diction function f˜in Theorem 17 are satisfied, we find Mf˜=M⁰f˜.

Proof: First, we prove infMf˜ ≥ 1 by contradiction. Assume there is γ satisfying kγk^Vh≥c₀>0 such that for any k <1(k∈R) hold

kE⁽¹⁾_h ( ˜f +h^kγ)k^Vh=O(h^κ1), κ₁≥κ≥6, kE⁽²⁾_h ( ˜f +h^kγ)k^Vh=O(h^κ2), κ₂≥κ₀.

4.5.1 Periodic cases 83 We denote E⁽¹⁾_h ( ˜f +h^kγ) =η and E⁽²⁾_h ( ˜f+h^kγ) =ζ, write E⁽¹⁾_h in the explicit form and obtain forn, n+ 1∈N_h,

γ(0,j) =h^−k(ζ −ˆr^(α0)), (4.65) Sγ(n+ 1) = (I+J^L)γ(n) +J^Q(γ(n),γ(n))

+J^Q(γ(n),2 ˜f(n)) +h^−k(η(n)−ˆr(n)), (4.66) in which E⁽¹⁾_h ( ˜f) = ˆr and E⁽²⁾_h ( ˜f) = ˆr^(α0) have been inserted. Carrying out a similar estimate as in section 4.4.1 (by replacingewith γ), we get a recursion inequality forγ (in the same form of (4.56)),

kγ(n+ 1)k²a,Ωh ≤ kγ(n)k²a,Ωh+h²αkγ(n)k²a,Ωh

βkγ(n)k⁴a,Ωh+h^−2k(2 + 1

h²)kη(n)−r(n)ˆ k²a,Ωh. (4.67) Further, we prove that there exists h₀ > 0 so that for all h < h₀ and for all n∈N_h

kγ(0)k²a,Ωh+

n−1

X

k=0

h^−2k(2 + 1

h²)kη(k)−r(k)ˆ k²a,Ωh ≤h²1

Te^−C0T. (4.68) Note that

kη−rˆk^Vh =O(h⁶+h^κ1),

kγ(0)k^Vh =kζ−ˆr^(α0)ka,Ωh=h^−kO(h^κ0 +h^κ2), the right hand side of (4.68) turns out to be

kγ(0)k²a,Ωh+

n−1

X

k=0

h^−2k(2 + 1

h²)kη(k)−ˆr(k)k²a,Ωh

=h²O(h^{2(κ0−1−k)}+h^{2(κ2−1−k)}) +h²O(h^6−2k+h^{2κ1−6−2k}) and is thus of sizeh²O(h^min(2(κ0−1−k),6−2k)). Sinceκ0 ≥α0+ 1≥2 and k <1, we can find someh₀ >0 to guarantee the inequality (4.68). Now Lemma 2 can be applied to recursion inequality (4.67) ofγ, which yields

kγk^Vh≤C^′h^κ0−k, nh² ≤T Taking the limith−→0, we have lim_h→0kγk^Vh=0.

Hence we achieve (−∞, 1)∩ Mf˜=∅. According to Proposition 2,Mf˜=M⁰f˜.

We check the possible elements ofMfˆand find, Lemma 9 If κ^∞₀ 6=∞, then κ^∞₀ ∈ Mf˜(E²_h).

Proof: For the sake of convenience, let k=κ^∞₀ . Further choose γ=f^∗ 6= 0 to be a constant function. Consequently kγk^Vh is constant too.

We insert ˆf +h^kγ into the update rule (2.20) and calculate the residue. Since the equilibrium function has a linear and a quadratic part, it turns out that

E⁽¹⁾_h ( ˆf +h^kγ) = E⁽¹⁾_h ( ˆf) +h^kA

f^L(γ) +f^Q(γ,2 ˆf +h^kγ)−γ

. But the last term on the right hand side is zero, because f^L(γ) = γ and h1,Vγi = 0 which cancels the term connected tof^Q. Hence E⁽¹⁾_h ( ˆf +h^kγ) = E⁽¹⁾_h ( ˆf).

Now, it follows that E⁽²⁾_h ( ˜f +h^kγ) = E⁽²⁾_h ( ˜f) +h^kγ. Further since

kE⁽²⁾_h ( ˜f +h^kγ)k^Vh ≤ kE⁽²⁾_h ( ˜f)k^Vh+h^kkγk^Vh, (4.69) thusord_E²( ˜f)⊆ord_E²( ˜f+h^kγ). In summary, we have proved the theorem.

At this point, it is not clear whether (k < κ^∞₀ ) ∈ Mfˆ(E²_h). However, from this theorem, we can conclude that ˆf may be consistent up to orderκ^∞₀ to the lattice Boltzmann scheme E²_h.

For the linear lattice Boltzmann scheme E²_h, we find that a standard conver-gence estimate is admitted in A , and combined with this theorem, we get the estimated convergence order of ˜f, which is a lower bound of the consistency order infMf˜.

For the nonlinear lattice Boltzmann scheme E²_h, we obtain a standard conver-gence estimate only in the subspaceA0. The estimated convergence order of ˜f is supposed to be the lower bound of infM⁰f˜. However, from Proposition 3 it follows that infM⁰f˜= infMf˜.

Summarizing this discussion, we achieve a conclusion about consistency order in periodic cases.

Theorem 20 (Consistency order of f˜ in periodic domains)

Let the linear part J^L of the collision operator in the lattice Boltzmann scheme E²_h admit the stability condition (4.14) and the stability estimate (4.15). Fur-ther let the additional condition (4.34) holds for the nonlinear E²_h. Assume f˜ satisfies the implications of Theorems 7, 8. Then

κ^∞₀ ∈ K0, (κ, κ^∞₀ )^T∈ord_E2( ˜f) (4.70) where

K0= (−∞, α₀+ 1]∪ {α₀+ 2, α₀+ 3, . . . ,5} ∪ {∞}, (4.71) and the consistency order is in the interval,

˜

σ= min(κ−2, κ^∞₀ )≤O˜_E²( ˜f)≤κ^∞₀ . (4.72)

4.5.1 Periodic cases 85 of the moments, the coefficients of ˆr^(α0) are uniformly bounded. This implies that the possible values ofκ₀can only be inK0. Henceκ^∞₀ ∈ K0 and (κ, κ^∞₀ )^T∈ ord_E²( ˜f) too.

Therefore we have a value ofκ₀ =κ^∞₀ . Since the estimated convergence order of f˜, which is min(κ−2, κ₀) = min(κ−2, κ^∞₀ ) (Theorem 11), cannot be larger than the consistency order according to Theorem 5. On the other hand, Theorem 9 tells us ˜O_E2( ˜f) = infMf˜is less or equal to κ^∞₀ . Thus (4.72) holds.

In addition, when κ−2 ≥ κ^∞₀ , we have min(κ−2, κ^∞₀ ) = κ^∞₀ . Hence (4.73) holds too.

Let us discuss several cases of (4.73).

In general, κ^∞₀ = α₀ + 1 (refer to the discussion following equation (3.93)).

Combined withκ−2≥4≥α₀+ 1, the convergence order and the consistency order are identical,α0+ 1. This implies that, in general, ˜f precisely represents the lattice Boltzmann solutions up to orderα₀+ 1.

In particular cases, for example the case in Remark 7, both κ₀ and κ^∞₀ are bigger than α0+ 1 when Eα0 (α0 = 1,2) is employed. Thus infMf˜ > α0+ 1.

When ψ(x) = 0 and ∂tG(0,x) = 0, ˆr⁽¹⁾(0) = ˆr⁽²⁾(0) = ˆr⁽³⁾(0). Hence κ₀ = 4 for all α₀, and maybe κ^∞₀ = 4 since f⁽⁴⁾(0,xj) 6= 0 generally, which implies again the convergence order and the consistency order are identical, and the regular expansion ˜f correctly stands for the lattice Boltzmann solution up to order 4.

In even more particular cases like Poiseuille flow in a horizontal channel, ˆr⁽³⁾ = 0 and κ₀ is ∞ if Eα0 (α₀ = 3) is applied. Therefore κ^∞₀ = ∞. Actually, in this case, the residue from the update rule ˆr(n,j) vanishes, since Poiseuille flow is a stationary, second order polynomial flow. Thereforeκ =∞, too. This implies that ˜f is the exact solution of the lattice Boltzmann method E²_h.

4.5.2 Cases with the bounce back rule Lemma 10 If κ^∞₀ 6=∞, then κ^∞₀ ∈ Mf˜(E³_h).

Proof: Again we choose γ=f^∗. Moreover,

E⁽³⁾_h ( ˜f +h^kγ) = E⁽³⁾_h ( ˜f)−h^kA(f^L(γ)−γ+f^Q(γ, h^kγ+ 2 ˜f)). (4.75) The remainder of the proof is similar to the case in Theorem 9.

Recalling the relationship between the estimated convergence order and the consistency order, we can discuss the consistency of ˜f more precisely for the linear schemes on bounded domains.

Proposition 4 (Consistency order of f˜on bounded domains)

Let the collision operator J^L in the linear lattice Boltzmann scheme E³_h ad-mit the stability condition (4.14) and the stability estimate (4.15). Assume f˜ satisfies the implications of Theorem 7, 9. Then O˜_E³( ˜f)∈[κb−3/2, κ^∞₀ ].

Proof: From Theorem 10, we knowκ^∞₀ ∈ Mf˜, thus infMf˜≤κ^∞₀ .

In addition ˜σ =κ_b−3/2≤infMf˜for the linear problem according to Theorem 5.

For the nonlinear problem E³_h, we still can get a upper bound from Theorem 10, infMf˜≤κ^∞₀ . However,M⁰_f˜=Mf˜is not obtained. Thus we only achieve κ_b−3/2 ≤infM⁰f˜ by applying Theorem 5. A lower bound for infMf˜ is not clear.

Im Dokument Analysis of Lattice Boltzmann Boundary Conditions (Seite 95-100)