Energy estimate and stability

Multiplying equation (3.2) by ǫ² and sending it to zero seems to avoid the trouble with infinite or undefined terms. However, under the hypothesis that f_ǫ behaves decently (e.g. staying bounded), the differential equation becomes in the limit

0 =J₀f(t, x) =hf,1iw−f,

which is dramatically different from (3.2). In contrast to the initial value problem having a unique solution, this purely algebraic limit equation permits an infinite multitude of solutions. Any function of the formf(t, x) =w(t, x)w, wherew:XT → Ris someL-periodic function respecting the initial conditionw(0, x) =v₀(x), solves the limit problem.

What may not be grasped at first glance is the fact that w is not at all arbitrary but the solution of (3.4). Clearly, this reveals the singular nature of the equation.

In order to disentangle the confusing situation, we will make the ansatz, that f_ǫ depends in a regular manner on ǫ. This means, that we try to approximate the solution by a truncated power series in ǫ. Amazingly, this procedure provides a lot of insight, although only a special class of initial values can be captured⁴.

3.1 Energy estimate and stability

Prior to the asymptotic consistency analysis, we establish a stability result. This states grosso modo, that the solution of the lattice-Boltzmann equation depends continuously on its initial value and a possible source term. The result is formulated in a way directly adapted to our purpose, guided by the following idea:

Consider an exact solution f_ǫ and an approximate solution ˆf_ǫ of the lattice-Boltz-mann equation. We can interpret the approximate solution as an exact solution with an additional source term corresponding to the residual. If stability holds, the two functions will stay close together during their evolution, provided they are initially almost equal and the residual is small. So asymptotic similarity can be inferred from stability, as soon as consistency is ensured, which means that the residual (i.e. the additional source driving ˆfǫ) vanishes withǫ.

A crucial role in the stability proof comes up to the following lemma which relies on the special structure of the equilibrium operator. In the sequel let W :F → F be the linear operator⁵ corresponding to a multiplication with the weight function w and letB =W⁻¹ its inverse.

Lemma 3.1. BJǫ is a symmetric (self-adjoint) operator with respect to the stan-dard scalar product in F. There exist orthogonal projectors Π_k, k∈ {1,2,3}, with mutually disjoint ranges such that BJ_ǫ permits the following representation:

BJ_ǫ =−P³

k=1

λ_kΠ_k=−P³

k=1

λ_kΠ^⊤_kΠ_k (3.5)

4It turns out that the zeroth order term of the expansion must be proportional tow. Therefore the initialization (3.1) has been chosen so restrictive. Since fǫ(0, x) ∈ F, we have fǫ(0, x) ∈ span{w,s,¹_θ −s²}. That is why the general initialization should be given byfǫ(0, x) = a(x)w+ b(x)s+c(x)`₁

θ −s²´

with some functionsa, b, c:X →R. Moreoverǫ-dependent initializations are imaginable, e.g. in the form of truncated power series.

5Iff= (f−,f0,f+)^⊤thenW = diag(_2θ¹,^θ−1_θ ,_2θ¹).

Here the −λ_k’s (withλ_k≥0) denote the non-positiveeigenvalues.

Proof: SinceBJ_ǫ =ω(BE_ǫ−B) and due to the symmetry ofB, the first part of the assertion is proved, if we verify, thatBEǫis symmetric too. Havingw=w·1=W·1, we find

BE_ǫ = (1−ǫ²τ c)h·,1i B W| {z }

=I

·1= (1−ǫ²τ c)h·,1i1.

So, BEǫ is just a multiple of the rank-one orthogonal projector _h1,1¹ _ih·,1i1, that is symmetric according to proposition 3.1. This implies that BEǫ has only a single non-zero eigenvalue given by 1−ǫ²τ c with the associated eigenvector is 1∈ F. By thespectral theoremforsymmetricoperators in Euclidean⁶vector spacesBJ_ǫhas an orthonormal basis of eigenvectors. Let Π_k fork∈ {1,2,3}denote theorthogonal projector⁷ on thek’th eigenvector. Since Π_j,Π_krefer to different eigenvectors ifj6= k we get Π_jΠ_k = 0. Using again proposition 3.1, the orthogonality of the projector Π_k entails its symmetry, i.e. Π^⊤_k = Π_k, and thus Π_k = Π²_k = Π^⊤_kΠ_k. Designating with−λ_k the eigenvalue ofBJ_ǫbelonging to Π_k, the spectral decomposition ofBJ_ǫ in (3.5) follows.

Let us finally check that all eigenvalues of BJ_ǫ are non-positive. Considering first the case c= 0, we find

BJ_ǫ =BJ₀ =ωB(h·,1iw−I) =ω h·,1i1−B

=ω

_{1 1 1}

1 1 1 1 1 1

−

_{2θ · ·}

· θ θ−1 ·

· · 2θ

Taking recourse to the explicit matrix representation of BJ₀, it can be verified that BJ₀ has the following non-positive eigenvalues:

0, −2ωθ, −ω(2θ+_θ−^θ₁ −3) withθ >1.

HenceBJ₀ is negative semi-definite. As the projector−ǫ²τ ch·,1i1 is negative semi-definite as well (recall τ >0 and c≥0) we conclude that BJ_ǫ =BJ₀−ǫ²τ ch·,1i1 is also negative semi-definite. Hence positive eigenvalues are excluded.

Proposition 3.1. Let P be a projector in a vector space V, i.e. P = P². Then the following two statements are equivalent:

i) P is orthogonal, i.e. ∀v, w∈V hP v,(I−P)wi=h(I−P)v, P wi= 0 ii) P is symmetric, i.e. P=P^⊤.

Proof:

i)⇒ii): Expanding the assumption yields: hP v, wi=hP v, P wi=hv, P wi. This implies: 0 =hP v, wi − hv, P wi=h(P−P^⊤)v, wi.

Since this holds for allv, w∈V, we conclude: P−P^⊤= 0 ⇔ P=P^⊤. ii)⇒i): From the hypothesis we infer: 0 =P−P²=P^⊤−P^⊤P =P^⊤(I−P).

Hence we find for allv, w∈V: 0 =hv, P^⊤(I−P)wi=hP v,(I−P)wi.

6By anEuclidean vector spacewe understand a finite-dimensional, real vector space equipped with a scalar product.

7Ifv6= 0 is a vector in an Euclidean vector space, then _hv,vi¹ h·, vivis theorthogonal projection onv.

3.1. Energy estimate and stability 131

Let us make the notion of an approximate solution more precise by recalling the definition ofconsistency. Anǫ-dependent function ˆf_ǫ ∈ Cper¹ (XT,F) is said to be con-sistent of orderα to the lattice-Boltzmann equation, if there exists anǫ-dependent functionr_ǫ∈ Cper(XT,F) – called the residue function– such that:

i) ˆfǫ is an (exact) solution of the lattice-Boltzmann equation with the additional source termǫ^αr_ǫ:

∂_tˆf_ǫ − ǫ⁻¹S∂_xˆf_ǫ = ǫ⁻²J_ǫˆf_ǫ + gw + ǫ^αr_ǫ. (3.6) ii) r_ǫ is uniformly bounded with respect toǫ:

∃C >0, ∀ǫ∈(0,1] :

Now we can formulate the central stability result of this chapter:

Theorem 3.1. Let f ∈ Cper¹ (XT;F) be a solution of the lattice-Boltzmann equation.

Suppose that ˆf_ǫ ∈ Cper¹ (XT;F) is consistent to the lattice-Boltzmann equation of order α >0 and

kf_ǫ(0,·)−ˆf_ǫ(0,·)k_L²(X;F)< K₀ǫ^α (3.8) for some constant K₀. Then there exists a constantK >0independent of ǫ∈(0,1]

such that

sup

t∈[0,T]kfǫ(t,·)−ˆfǫ(t,·)kL²(X;F)< Kǫ^α .

Proof: For convenience let us introduce thedeviationd_ǫ :=f_ǫ−ˆf_ǫ. As our equilib-rium operator is linear, the evolution equation ford_ǫis simply derived by subtracting (3.6) from (3.2).

∂_td_ǫ+¹_ǫS∂_xd_ǫ= _ǫ¹2 J_ǫd_ǫ−ǫ^αr_ǫ (3.9) Moreover note, that B is a positive operator in F, whose operator norm satisfies 1≤min{2θ,_θ−^θ₁} ≤ kBk ≤max{2θ,_θ−^θ₁}ifθ >1 andkBk= 2 forθ= 1. Therefore the norm defined by ||| · |||²_L²_(X;F) :=R

XhB·,·i is equivalent to the standard norm in L²(X;F) and we have k·k²_L²_(X;F)≤ ||| · |||²_L²_(X;F) ≤ kBk k·k²_L²_(X;F).

Now the proof is done by an energy estimate of the deviation. This is obtained by taking the scalar product of equation (3.9) with Bd_ǫ and integrating over the spatial domain X. Let us consider the second integral on the left hand side, which is given in more detail by:

Since f_ǫ,ˆf_ǫ and consequently d_ǫ are L-periodic, i.e. d_ǫ(t,0) = d_ǫ(t, L), the integral vanishes. Remarking that hBd_ǫ, J_ǫd_ǫi = hd_ǫ, BJ_ǫd_ǫi, because B is diagonal as a multiplication operator and a fortiori symmetric, we can exploit lemma 3.1. So equation (3.10) becomes

The expression resulting from the integral over the collision term (the first one of the right hand side) is obviously negative. Thus we can estimate applying additionally the Cauchy-Schwarz inequality: for arbitrary reals a, b, ǫ. Finally, we apply Gronwall’s inequality⁸:

|||dǫ(t,·)|||²_L²(X;F) ≤ exp(t)|||dǫ(0, .)|||²_L²(X;F)+ǫ^2αexp(t)kBk Z t

0 krǫ(σ,·)k²_L²(X;F)dσ By the initial condition (3.8) we have|||d_ǫ(0,·)|||_L²(X;F)≤ kBkK₀ǫ^α. From the defini-tion of consistency we know, that the integral is dominated by some constantC >0 for 0≤t≤T. So we end up with

Due to its linearity and the constancy of the parameters ω and c, its form will

8Gronwall’s lemma in differential form claims, that the non-negative function η ∈ C¹([0, T]) satisfies the estimate

if the differential inequality _dt^dη(t) ≤ φ(t)η(t) +ψ(t) holds true with the nonnegative functions φ, ψ∈ L¹([0, T]). For further reference see [17] page 624.

Im Dokument Analysis of Lattice-Boltzmann Methods : Asymptotic and Numeric Investigation of a Singularly Perturbed System (Seite 141-145)