Consistency and asymptotic similarity

It has been shown in the preceding section by a formal computation, that the lattice-Boltzmann scheme is related to the diffusion-advection-reaction equation.

Classically, a finite difference scheme is connected to a differential equation by a consistency analysis, where the solution of the differential equation is plugged into the scheme. Under suitable smoothness assumptions, the finite differences are replaced by truncated Taylor expansions. If the solution of the differential equation satisfies the difference scheme up to a remainder that shrinks to zero for the discretization getting finer and finer, the scheme is called consistent to the differential equation. Then the output of the scheme is considered to approximate the solution of the differential equation.

In order to prove this approximation in a rigorous way, a stability discussion is nec-essary. For this the scheme is associated with a linear operator acting on a space, whose dimension is determined by the chosen grid. Stability means, that the family of operators resulting from global grid refinements is bounded independently of the mesh parametrized by the grid spacing. By virtue of theLax-Richtmyer equivalence theoremconvergence of a linear scheme is guaranteed as soon as it is consistent and stable.

However, this standard approach – as just outlined – is not directly applicable to the analysis of lattice-Boltzmann schemes, as it involves more variables (unknowns) with the tuple-valued population function than the approximated target equation. So the consistency analysis cannot be performed in the manner described above, since it is not possible to insert the solution of the diffusion-advection-reaction equation into the lattice-Boltzmann algorithm. Nevertheless, we may follow the guideline of the classical analyzing procedure, if we generalize the notion of consistency.

The lattice-Boltzmann algorithm in operator form and further notation.

In the sequel we consider the lattice-Boltzmann algorithm as described on page 160 with a terminal time 0< T <∞. We recall that N_h denotes the number of spatial nodes forh∈ HL, whereHL is the set of admissible grid spacings.

Basically, the lattice-Boltzmann algorithm assigns to a given function corresponding to the state at time-stepn−1, a new function corresponding to the state at time-step n. So, we can describe the algorithm on the grid Gh by an evolution operator,

B_h(n) :F(S ×Z/N_hZ,R)→ F(S ×Z/N_hZ,R) n∈T_h\ {0}

where the argument nindicates, thatB_h might explicitly depend on the iteration, if for instance the external source depends on time. Since the equilibrium depends linearly on the zeroth moment U_h, the evolution operator is linear as long as there are no external sources. In general, however, the evolution operator is an affine-linear mapping. Equivalently, the lattice-Boltzmann equation (4.5) can now be written in operator form,

F_h(n+ 1,·) = B_h(n+ 1)F_h(n,·) = E_hF_h(n,·) + G_h(n,·) where the linear part is given by the evolution matrix

E_h :F(S ×Z/N_hZ,R)→ F(S ×Z/N_hZ,R).

4.2. Consistency and asymptotic similarity 173

Supposing that the quantities τ, a, candω=τ⁻¹ are constant, E_h does not depend on the iteration. Let E_h,s F_h(n,·), i

denote the evaluation of E_hF_h(n,·)∈ F(S × Z/N_hZ) for s∈ S and i∈Z/N_hZ. Then the evolution operator is defined by

E_h,s F_h(n,·), i

:= (1−ω)F_h,s(n, i−s) + Eh,s

X

˜ s∈S

F_h,˜s(n, i−s) . The additive part, containing the external source g, is given by the grid function G_h :T_h→ F(S ×Z/N_hZ,R).

G_h,s(n, i) := h²g(t_n, x_i)w_s The restriction operator⁸

R_h :F([0, T]×R/LR,R)→ F(T_h×Z/N_hZ,R)

confines every function f ∈ F([0, T]×R/LR,R) to the time-space grid Gh. If R_h(f, n, i) denotes the evaluation ofR_hf at the time-step n∈T_h and at the mesh-node i∈Z/N_hZ, thenR_h is defined by

R_h(f, n, i) :=f(tn, xi) with (tn, xi)∈ Gh.

We will mostly use the short notation ˆf(n, i) :=R_h(f, n, i); it should then become clear from the context, to what grid the restriction refers.

Let us finally equip the function space F(S ×Z/NZ,R) with some norms⁹. Since dimF(S ×Z/NZ,R) = #S · #(Z/NZ) = 3N ,

the space is isomorphic toR^3N. Therefore we obtain canonically the norms:

kfk∞ := max

s∈S max

i∈Z/NZ|f_s(i)| kfkp := X

s∈S

X

i∈Z/NZ

|f_s(i)|^p1_p

p≥1

As we interpret the functions of F(S ×Z/NZ,R) not as elements of R^3N but as restrictions of functions inF(S ×R/LR,R), we prefer¹⁰ theL^p-norms.

kfk^Lp := ^p qL

Nkfk^p = X

s∈S

X

i∈Z/NZ

|fs(i)|^{p L}_N¹_p

In order to adopt a uniform notation, we write k · kL^∞ instead of k · k∞. If not indicated otherwise, we suppose 1≤ p≤ ∞. It should be remarked, that L^p(S × Z/NZ,R) is isometrically embeddedintoL^p(S ×R/LR,R) by identifyingi∈Z/NZ withx_i:=i_N^L ∈R/LR. The functions which are thus defined only on the nodesx_i

8By abuse of notation, we utilize the same symbol also for the restriction operator onF(S × [0, T]×R/LR,R) that is defined analogously.

9It should be clear, that everything what follows has a straight-forward analogon inF(Z/NZ,R).

However, we do not state the analogous definitions, to avoid repetitions.

10Note that for the constant one-function 1N ∈ F(Z/NZ,R) we have k1Nkp= √^p

N, while we havek1NkL^p= √^p

Lbeing independent ofN. In that sense, theL^pnorms are better to be compared with each other, ifN varies.

are extended by the left neighbor node interpolation (or alternatively by the nearest neighbor node interpolation) to piecewise constant functions defined on the whole domain R/LR.

L^p(S ×Z/NZ,R)∋H7→H¯ ∈L^p(S ×R/LR,R)

Left neighbor node interpolation: H¯_s(x) :=







H_s(x₀) x₀ ≤x <x₁ ... ... ≤x < ... H_s(x_N−1) x_N−1≤x < L Note, that the L^p-norm ofH is equal to the standardL^p-norm of the tuple-valued function ¯H onR/LR, i.e. kHkL^p =kH¯kL^p := P

s∈S

R

R/LR|H¯_s(x)|^pdx¹_p .

Definition 4.1. (Consistency) Let f_ǫ ∈ F(S ×[0, T]×R/LR,R) depend on the real parameter ǫ. The function f_ǫ is consistent to the discrete lattice-Boltzmann equation with a consistency orderr ∈R⁺₀ referring to the normk·kL^p, if there exists a constant C_p independent of h and n, such that for all h∈ HL and n∈T_h\ {0} the iteration error satisfies the estimate

ˆf_h(n,·)−B_h(n)ˆf_h(n−1,·)

L^p ≤ C_ph^r . (4.24) The reason why we are interested in a consistent population functionf_ǫis to compare it for ǫ=h with the outputF_h of the lattice-Boltzmann algorithm on the gridGh. Since the solution of the discrete lattice-Boltzmann equation is uniquely defined by the initial values and since ˆf_h solves the discrete lattice-Boltzmann equation approximately, we expect, thatF_h and ˆf_h remain close together if they are (almost) equally initialized. Of course, the deviation might increase in time mainly because ˆf_h does not satisfy the discrete lattice-Boltzmann equation exactly. Let us suppose, that the deviation per iteration is of order O(h^r). In order to reach a fixed time t ∈ (0, T], N_t =N_t(h) := [tN²] = [th⁻²] iterations have to be performed. In the worst case, the deviation per iteration cumulates; thus after [th⁻²] iterations the mutual deviation of F_h and ˆf_h should be of order [th⁻²]O(h^r) = O(h^r−2), under the assumption that both quantities agree at the beginning.

Before we go on, let us mention the following lemma, that turns out to be quite helpful in the proof of the next theorem.

Lemma 4.3. For k ∈ N consider the affine-linear mappings A_k : V → V with V denoting some vector space. Assume A_kx = L_kx+b_k with the linear mapping L_k∈Hom(V) and some vector b_k∈V. Then for x, y∈V

Qn k=1

A_kx − Qn k=1

A_ky = Qn k=1

L_k(x−y) (4.25)

where the product sign denotes the composition, for instance Q³

k=1

A_kx= (A₃◦A₂◦ A₁)(x) =A₃

A₂ A₁(x) .

Proof: The proof is done by induction:

4.2. Consistency and asymptotic similarity 175

The following theorem is related to the Lax-Richtmyer equivalence theorem, which plays a central role in the theory of linear finite difference schemes. Actually, this theorem comes along in various versions depending on the difference scheme to be considered.

Generally, it states, that a linear finite difference scheme is convergent if and only if it is consistent and stable, i.e. consistency and stability are sufficient and necessary conditions for convergence. The power of the equivalence theorem is based on the fact, that it permits to divide the convergence analysis into two parts that can be done independently: a consistency analysis that is often quite evident by a Taylor expansion and a stability analysis.

In the following theorem we provide the central estimate that connects – under the hypothesis of stability – consistency with asymptotic similarity, thus paving the way towards theconvergence of the considered D1P3 lattice-Boltzmann scheme.

Theorem 4.1. May F_h for h ∈ HL be a solution of the discrete lattice-Boltzmann equation (4.5) initialized by F_h(0,·) and may f_ǫ be consistent of order r to the discrete lattice-Boltzmann equation with respect to theL^p-norm. Then the difference between F_h andˆf_h can be estimated by

The mutual deviation is obviously split into two parts: Theinherited initial devia-tion amplified or attenuated by the repeated application of the evolution operator and theinherent deviation, that is due to the iteration error.

Proof: Let us introduce the two grid functions A_h(·) := ^NQ^t

Since F_h(N_t,·) = ^NQ^t

k=1

B_h(k)F_h(0,·), we get by using the triangle inequality:

F_h(N_t,·)−ˆf_h(N_t,·) ≤ kA_hk+kB_hk

So, we have to estimate A_h and B_h. Application of lemma 4.3 yields:

A_h = ^NQ^t

B_h can be written as a telescopic sum in the following manner.

B_h =

Exploiting once more lemma 4.3 gives:

=

hstands for the identity. Using now the consistency (4.24) and observ-ing that Nt≤tN² =th⁻², this results in the following estimate:

Hence, the difference between the lattice-Boltzmann solutionF_h and the given

func-tion ˆf_h is dominated as asserted.

It is desirable to improve estimate (4.26) in the sense that the h-dependence of the right hand side gets more simple. In order to finish the conclusions, that the preceding theorem permits us to draw, we adopt the following assumption of stability:

sup

h∈HL

0≤nmax≤[th⁻²]kEⁿ

hkp =:E_p <∞ (4.27) Furthermore let us suppose, that F_h and ˆf_h areinitially consistentof orderρ(w.r.t.

the L^p-norm), meaning that there exists a constantJ_p >0 being independent ofh with

kF_h(0,·)−ˆf_h(0,·)kL^p ≤ J_p h^ρ. (4.28)

4.2. Consistency and asymptotic similarity 177

Then (4.26) becomes:

kF_h(N_t,·)−ˆf_h(N_t,·)kL^p ≤ E_pJ_p h^ρ + E_pC_pT h^r−2 (4.29) As both quantities on the left hand side depend on h, it does not make sense to speak aboutconvergenceofF_h(N_t(h),·) towards ˆf_h(N_t(h),·) or conversely forh↓0.

Instead, we refer to definition 1.4 where we have introduced the notion ofasymptotic similarity, characterizing this kind of mutual approximation.

Corollary 4.1. May F_h for h∈ HL be a solution of the discrete lattice-Boltzmann equation (4.5) and mayf_ǫ be consistent of orderr to the discrete lattice-Boltzmann equation w.r.t. the normk · kL^p. The stability assumption may be fulfilled andF_h,ˆf_h are supposed to be consistently initialized up to orderr−2. ThenF_h(N_t,·),ˆf_h(N_t,·)∈ L^p(S ×Z/N_hZ,R) ^isom.֒→ L^p(S ×R/LR,R) are asymptotically similar as sequences in L^p(S ×R/LR,R) for h↓0 with residue of order r−2, as we have:

kF_h(Nt,·)−ˆf_h(Nt,·)k^Lp ≤ (EpJp + EpCpT) h^r−2 (4.30) Proof: The assertion is a direct consequence of the isometry and equation (4.30),

that results trivially from (4.29).

Eventually, we throw a glance at the zeroth moment, which is actually the quantity of interest. Let 1≤p <∞.

In order to get the summation over s∈ S out of the modulus, we remember that x7→ |x|^p is a convex function¹¹ forp≥1. Thus we obtain: that the asymptotic similarity is transmitted from the population functions F_h,ˆf_h to the associated zero momentsU_h,uˆ_h.

11If the functionfis convex on the intervalI ⊂R, then the following inequality holds true for npointsx1, ..., xn∈Iand positive numbersλ1, ..., λn withλ1+...+λn= 1:

f(λ1x1+...+λnxn)≤λ1f(x1) +...+λnf(xn) Therefore we obtain in our special case:

|P

Im Dokument Analysis of Lattice-Boltzmann Methods : Asymptotic and Numeric Investigation of a Singularly Perturbed System (Seite 184-190)