Definition of consistency order

0 0.5 1

0.015 0.02 0.025 0.03 0.035 0.04

−1 −0.5 0 0.5 1

−0.04

−0.02 0 0.02 0.04

Figure 3.12: Left: plot of (ˆu−hu)/(h²) for the linear flow (3.83) along the cuts x= 0.1,0.3. Right: plot of (ˆu−hu)/(h²) (∗) and (ˆv−hv)/(h²) (◦) at t= 0.6 for the circular flow. h= 0.02 for both cases.

0 0.5 1

1.8 2 2.2 2.4 2.6 2.8

3x 10⁻³

0 0.5 1

−3

−2

−1 0 1 2 3x 10⁻³

Figure 3.13: Plot of (ˆu−hu)/(h²) along several cuts for the decaying Taylor vortex flow. Left: along the cuts y = 0.1 (◦), 0.3 (∗). Right: along the cuts x= 0.1 (◦), 0.3 (∗).

3.7 Definition of consistency order

After analyzing the residue of the initial conditions and boundary conditions, we can discuss the consistency order of the numerical scheme more carefully.

We remark that, usually, the consistency order is defined based on the order of the residue in such a way, that it equals the convergence order of the method.

To follow this convention we assume a certain type of convergence estimate depending on the various residues corresponding to the different parts of the algorithm. Based on the estimate, we define upper bounds for the convergence order. The general expectation is that the least upper bound is a reasonable order of consistency.

As we have always stated, the lattice Boltzmann method is considered to solve a

problem such as (2.1). Moreover, from the asymptotic analysis, we have defined a truncated expansion (3.65) which shows a clear relation to the Navier-Stokes solution. If we have the consistency order of the expansion (3.65) with respect to the lattice Boltzmann solution f, we can conclude how well this expansion represents the lattice Boltzmann solutions and thus clarify the relation to the Navier-Stokes equation.

For the mathematical setting of our discussion, we first introduce the discretiza-tionNhof the time interval [0, T] and Ωhof the spatial domain Ω for an arbitrary gird sizeh∈H, whereH is the set of all the discretization parameters possibly used in the simulations, which contains an accumulation point at zero. Next, we introduce a linear and normed grid function space V_h equipped with norm k·k^Vh. The lattice Boltzmann solutionf and the truncated expansion ˆf on Ω_h are two functions in V_h. Furthermore, we define a grid function set

A=n s

s(h)∈V_h, h∈Ho

in which each element is a function ofh and grid nodes in Ωh and contains, for example, f and ˆf. Our discussion is build onAθ which may be some subspace of AorA itself.

In addition, we denote a complete lattice Boltzmann algorithm by a set of mappings,

E⁽ⁱ⁾_h : V_h →V_h, i= 1, . . . , m, (3.113) where f is the solution of

E⁽ⁱ⁾_h (f) = 0, i= 1, . . . , m, (3.114) Generally we write a scheme in a concise way by E^m_h or E^m, which contains m mappings representing the inner algorithm, the initialization algorithm, etc.

For any s∈ A, E⁽ⁱ⁾_h (s) is the residue with respect to the ith mapping.

The structure of an arbitrarily grid functions∈A_θ is analyzed at any possible order k∈Rof h by checking the residue of modificationss+h^kγ with γ∈ A. With the help of the following set

ord_E^m(s) =n

κ∈R^m

kE⁽ⁱ⁾_h (s)k^Vh=O(h^κi), i= 1, . . . , m.o

, (3.115) which contains the powers of all possible order estimate of the residues ofs, we have the following definition

Definition 1 (Undetermined order of s in Aθ)

Given a lattice Boltzmann algorithm E^m_h and s ∈ Aθ. If there exists γ ∈ A which satisfies c₁ ≥ kγk^Vh ≥c₀ >0 for suitable constants and all h∈H such that s+h^kγ ∈ Aθ and

ord_E^m(s)⊆ord_E^m(s+h^kγ)

then we say that k is an undetermined order of s in Aθ under this lattice Boltzmann scheme.

3.7. DEFINITION OF CONSISTENCY ORDER 57 A direct result of this definition is

Lemma 1 Let E^m_h be a lattice Boltzmann algorithm and s∈ A^θ. If s has an undetermined order k in Aθ under E^m_h, then k is also an undetermined order of sin A.

Intuitively ifkis an undetermined order ofsin the spaceA^θ, it is already clear thatswill generally not describe the numerical solution correctly in that order.

To make this more clear, we first introduce a general notion of convergence with the help of setord_E^m(s).

Definition 2 (A standard convergence estimate in Aθ)

Let f be the solution of the lattice Boltzmann algorithm E^m_h. We say that E^m_h admits a standard convergence estimate in Aθ, if there exists a monotone increasing function σ : R^m → R such that for any s ∈ Aθ and any κ ∈ ord_E^m(s) holds kf −sk^Vh =O(h^σ(κ)). We call σ(κ) an estimated convergence order of s.

Based on the above definitions, we have the following conclusion.

Theorem 5 Let f be the solution of the lattice Boltzmann scheme E^m_h, which possesses a standard convergence estimate in Aθ based on the monotone in-creasing function σ : R^m → R. Assume that s ∈ A^θ has an undetermined order k in Aθ. Then σ(κ)≤k for any κ∈ord_E^m(s).

Proof: (By contradiction).

Assume there is κ∈ordE^m(s) such thatσ(κ)> k.

Due to the standard convergence estimate inAθ,kf−sk^Vh =O(h^σ(κ)). Hence there is a constant c₁ and some h₁ > 0 such that kf −sk^Vh ≤ c₁h^σ(κ) for h < h₁. Since s is not determined in order k under the scheme E^m_h, there is γ∈ Asuch thats+h^kγ∈ Aθ and

ord_E^m(s)⊆ord_E^m(s+h^kγ)

Henceκ∈ordE^m(s+h^kγ), and there are constants h2,c2 such that

kf −s−h^kγk^Vh ≤c₂h^σ(κ) (3.116) holds for any h < h2. Neither c1 nor c2 depend on h. Since for any h <

min(h₁, h₂),

kh^kγk^Vh =k(h^kγ+s−f) + (f −s)k^Vh

≤ k(h^kγ+s−f)k^Vh+k(f−s)k^Vh≤(c₁+c₂)h^σ(κ), we have

0<kγk^Vh ≤(c₁+c₂)h^σ(κ)−k.

Taking the limit h −→ 0 of the right side, we obtain lim_h→0kγk^Vh = 0. This contradicts withkγk^Vh> c₀>0. Thereforeσ(κ)≤k.

This little observation gives us a hint how to define the consistency order of s which is quite likely to be equal to the best possible convergence order, if the scheme is convergent in A. Knowing that the estimated convergence order is not bigger than the smallest undetermined order, we define

M^s(E^m_h) :={k∈R |sis undetermined by E^m_h at order k inA.}, M^θs(E^m_h) :={k∈R|sis undetermined by E^m_h at order kinAθ.}, to collect all the undetermined orders of sinAand Aθ respectively.

Definition 3 (Consistency order of s w.r.t. the LB algorithm) The consistency order O˜_E^m(s) of sis equal to infMs(E^m_h).

As a direct conclusion of Lemma 1, we have

Proposition 1 For a given lattice Boltzmann algorithm and s∈ Aθ, holds M^θs⊂ Ms, infMs≤infM^θs.

Usually, computing the consistency order is not a simple task due to a lack of uniqueness and existence results for the equations governing γ. However, once we know some k ∈ Ms(E^m_h), we can conclude that the consistency order of s is not larger than k. On the other hand, the estimated convergence order may help to get the correct consistency order by means of Theorem 5. Even in the case that the standard convergence estimate of the scheme is only available in a subspaceAθ, the consistency order ofsmay still be obtained (this conclusion for ˆf will be verified in chapter 4).

Chapter 4

Convergence of the lattice Boltzmann method

In this chapter we intend to study the convergence of the lattice Boltzmann method introduced in chapter 2. Our considerations are a direct continuation of the stability result [45] which provides suitable norm estimates for the solu-tion of thelinearlattice Boltzmann method both in the case of periodic domains and on bounded domains in connection with the bounce back rule. The gen-eral structure of our convergence proof follows the approach in [73]: stability combined with an asymptotic expansion of the numerical solution yields con-vergence. This idea has been successfully applied to the convergence theory [18, 19] for nonlinear convective-diffusive lattice Boltzmann methods.

Recalling that the nonlinear term in the presented lattice Boltzmann algorithms appears only in the collision part and the collision operator can be divided into a linear and a quadratic part, i.e.,J(f) = J^L(f) +J^Q(f,f), we stress that a careful analysis of the linear case is an important step towards a convergence proof of the lattice Boltzmann method with non-linear equilibrium functions.

In fact, for a continuous lattice Boltzmann equation a rigorous convergence proof in the nonlinear case could be obtained [46] by controlling only the linear part J^L of the collision operator. Our convergence proof is therefore carried out also in two parts: linear and nonlinear.

This chapter begins with a review of the stability results from [45], then in section 4.2 the truncated asymptotic expansion of linear and nonlinear lattice Boltzmann solutions is recalled. With the help of stability results [45] and truncated expansion, we then give a rigorous convergence proof for both the linear (section 4.3) and the nonlinear (section 4.4) lattice Boltzmann methods.

4.1 Stability of the linear BM

The stability result presented in [45] is formulated for lattice Boltzmann al-gorithms with linear collision operator J^L on periodic domain or on bounded domains with bounce back rule at the boundaries. In order to express the norm estimates in a compact form, the lattice Boltzmann algorithm is formulated in

59

evolution operator notation. The relation to the abbreviation E⁽¹⁾_h , E⁽²⁾_h , E⁽³⁾_h , which is introduced in chapter 3, is also presented.

4.1.1 Function space

In this work, the lattice Boltzmann algorithms are set up by discretizing the time interval [0, T] (T > 0) into a set of time points tn =nh² and the spatial domain Ω into a node set of xj =jh. We denote the related index sets by

Nh=h⁻²[0, T]∩Z, Ωh =h⁻¹Ω∩Z^d (4.1) for an arbitrary gird size h. Based on these two sets, we define a grid function space

G^h ={s : Ωh →R^N}, I :G^h → G^h is the identity operator. Moreover, we set

V_h=n s

s : N_h×Ω_h →R^No

, (4.2)

in this chapter andAis based on this set according to the discussion in section 3.7. The set H of grid parameters will be given differently with respect to the different algorithms. Obviously A contains all grid functions in V_h. For an element s ∈ V_h, we denote s(n) ∈ Gh for any n ∈ Nh. Later a discrete integration norm is introduced on G^h, and the norm k·k^Vh is related to the temporal maximum of this discrete integration norm.

It is emphasized that for a given grid function s∈ A, the argument h is often suppressed without causing conflicts like the lattice Boltzmann solution f and the previously obtained regular expansion ˆf.

As for the lattice Boltzmann algorithm E^m_h, we let the second mapping in a scheme E^m_h correspond to the initial condition by setting

E⁽²⁾_h (s)(0) = Eα0(s),

E⁽²⁾_h (s)(n) = 0, n6= 0, n∈N_h, (4.3) in which Eα₀ is the initial condition introduced in Theorem 3 associated with I^(α0),

Eα₀(s) =s(0,j)− I^(α0)(x_j). (4.4) The lattice Boltzmann solution satisfies E⁽²⁾_h (f) = 0.

Apparently E⁽²⁾_h is required for problems on both periodic domains and general bounded domains. The other members of E^m_h will be given explicitly for the periodic case and bounded case with the bounce back rule in the next two sections.

4.1.2 Reformulation of lattice Boltzmann algorithms on periodic domains 61 4.1.2 Reformulation of lattice Boltzmann algorithms on

peri-odic domains

In section 2.4.1, we have described the case of periodic domains, and have rewritten the lattice Boltzmann algorithm with the help of modulo addition.

In particular, the grid size is chosen to be in the set

H={1/m |m∈N} (4.5)

in order to implement the periodicity condition exactly on the boundary.

Here we take up this description and further define the associated shift operator (Sf)i(j) =fi(j+mci), i= 1, . . . , N, x_j ∈Ω. (4.6) Then we can formulate the lattice Boltzmann algorithm (2.20) in the compact form

Sf(n+ 1) =f(n) +J(f(n)) +g(n).

Heref(n), g(n)∈ Gh are the vectorsf(n,j) andg(n,j) (j∈Ω_h) respectively.

In addition, we let E⁽¹⁾_h be the update rule and complement it with zero equa-tions atn= 0,

E⁽¹⁾_h (s)(n) =Ss(n)−[s(n−1) +J(s(n−1)) +g(n−1)], n∈Nh\ {0}, E⁽¹⁾_h (s)(0) = 0.

(4.7) Obviously,f satisfies

E⁽¹⁾_h (f) = 0. (4.8)

Combined with the initial mapping E⁽²⁾_h , a complete lattice Boltzmann scheme is obtained to solve a periodic problem.

4.1.3 Reformulation of lattice Boltzmann algorithms with the bounce back rule

In section 2.4.2, the bounce back rule has been described, and is applied at every boundary nodex_j to the components off which belong to incoming velocities.

On the other hand, for each velocity ck ∈ V, a boundary node set can be introduced, in which each node possesses the incoming direction c_k∗ = −c_k, i.e.,

∂kΩ ={x_j :x_j+ck =x_j+hck∈/ Ω}.

Using the new index symbol, the bounce back rule atxj ∈∂kΩ has the form f_k∗(n+ 1,j) =f_k(n,j) +J_k(f(n,j)) +g_k(n,j) + 2hF_k^(eq)_∗ (0,φ(tn,x_jk)) wherexjk is the intersection point of the link alongck and the boundary∂Ω.

Again, the full lattice Boltzmann scheme can be written in terms of a suitable shift operator ˜S :Gh → Gh in the compact form

Sf˜ (n+ 1) =f(n) +J(f(n)) +g(n) +b(n), (4.9) where

( ˜Sf)k(j) =

(fk(j+c_k) x_j ∈Ω\∂kΩ fk^∗(j) xj ∈∂kΩ and b(n)∈ Gh is defined by

b_k(n,j) =

(0 x_j ∈Ω\∂kΩ, 2hF_k^(eq)_∗ (0,φ(t_n,x_jk)) x_j ∈∂_kΩ.

We still use E⁽¹⁾_h to denote the update rule, and let E⁽³⁾_h correspond to the bounce back rule. However, the update rule is implemented only for the non-incoming velocities in V at all nodes, and the bounce back rule only for the incoming directions at boundary nodes. We define trivial equations for the missing directions,

(E⁽¹⁾_h (s))_k(n,j) = 0, x_j ∈∂_kΩ, n∈N_h\ {0}, k= 1, . . . , N, (E⁽³⁾_h (s))k(n,j) = 0, x_j ∈Ω\∂kΩ, n∈Nh\ {0}, k= 1, . . . , N, E⁽¹⁾_h (s)(0,j) = 0, j∈Ωh,

E⁽³⁾_h (s)(0,j) = 0, j∈Ω_h.

Hence the scheme (4.9) can be translated into

E⁽¹⁾_h (s)(n+ 1) + E⁽³⁾_h (s)(n+ 1) = ˜Ss(n+ 1)−[s(n) +J(s(n)) +g(n) +b(n)]

(4.10) for all n, n+ 1∈Nh and the lattice Boltzmann solutionf satisfies

E⁽¹⁾_h (f) + E⁽³⁾_h (f) = 0. (4.11) In addition, the set of discretization parameters

H ⊂(0, 1], (4.12)

contains an accumulation point at zero. LetδΩ_h ⊂Ω_hbe the set of all boundary nodes for which at least one neighbor is missing, then

δΩ_h =∪^Ni=k∂_kΩ.

Particularly, in the convergence proof, we require a property of the number of boundary nodes |δΩh|

h^d|δΩh| ≤ηh (4.13)

for someh-independent constantη >0 which expresses the fact that the bound-ary is one dimension lower than the volume.

4.1.4 Norms and stability 63 4.1.4 Norms and stability

The key point in the stability analysis is the definition of the stability structure (P,a, λ) consisting of a matrixP ∈R^N×N of left eigenvectors ofJ^L correspond-ing to the eigenvalues −λ₁, . . . ,−λ_N and suitable weights a_i > 0 collected in The first condition is the well known necessary stability condition in the linear case [67]. The second condition states that the rows of P contain the left eigenvectors of J^L. The third condition is an additional requirement on the eigenvectors ofJ^L, which can be reformulated as an orthonormality condition with respect to the weighted scalar product (for details see [45])

hf,gi∗ =

It is required for proving that both the shift operator and the operator (I+J^L) on the right hand side of the lattice Boltzmann equation (see (4.8), for example) have norms bounded by one. I is the identity operator.

A suitable norm for the stability estimate is defined in terms of a weighted inner product based on the vector a

hf,gia=

Summing over all nodes of the grid, we obtain a discrete integral norm onG^h for grid functions

kfk²a,Ωh= X

j∈Ωh

h^dkf(j)k²a, f ∈ Gh.

Based on it, a norm for convergence and consistency study is defined onV_h, kfk^Vh= max

n∈Nhkf(n)ka,Ωh, f ∈V_h. The operator norm associated tokka,Ωh is given by

kBka,Ωh = sup

f∈Gh\{0}

kBfka,Ωh

kfka,Ωh

.

Similarly, for matrices C∈R^N×N, we define kCka= sup

f∈RN\{0}

kCfk^a kfka

.

With these notations, we can state the stability result presented in [45].

Theorem 6 If the lattice Boltzmann model possesses a stability structure(P,a,λ) with a symmetric weight vector (i.e. a_i =a_i∗) satisfying (4.14), then S and S˜ are isometries and

kSka,Ωh= 1, kS⁻¹ka,Ωh= 1, kS˜ka,Ωh = 1, kS˜⁻¹ka,Ωh = 1, (4.15) kI+J^Lka≤1, kI +J^Lka,Ωh≤1. (4.16) It is also shown in [45] that the standard BGK, TRT, and MRT models possess the required stability structure.

We conclude this summary with a remark on how to estimate the conserved moments ρ(f) =P

ifi and u(f) =P

ifici. Recalling that the first row of P is 1^T and the nextdrows are v^T₁, . . . ,v^T_d, we can also writeρ(f) = (Pf)₁ and uα(f) = (P f)1+α. Now, in view of the third condition in (4.14)

kPfk² =hPf, Pfi=hP^TPf,fi=kfk²a, so that

|ρ(f)|=|(Pf)₁| ≤ kPfk=kfk^a, ku(f)k=q

(Pf)²₂+· · ·+ (Pf)²_d+1 ≤ kPfk=kfka. In particular, we obtain for the discrete spatial L² norms

kρ(f)k²Ωh = X

j∈Ωh

h^dρ(f(j))²≤ kfk²a,Ωh, ku(f)k²Ωh = X

j∈Ωh

h^dku(f(j))k² ≤ kfk²a,Ωh. (4.17) and for a discrete maximal norm

ku(f)k∞,Ωh= max

n∈Nhku(f(n))kΩh. (4.18)

4.2 The truncated asymptotic expansion and

Im Dokument Analysis of Lattice Boltzmann Boundary Conditions (Seite 69-78)