Initial conditions and irregular expansions

Corollary 3.4. (Convergence of Moments) Under the conditions of the preceding theorem the mass moment u_ǫ =hf_ǫ,1i converges quadratically inǫto the solution v of (3.32), while the first and second moment φǫ, ψǫ vanish in the limit. However, the the limit of the rescaled moments ǫ⁻¹φ_ǫ and ǫ⁻²ψ_ǫ is related to the first and second derivative of v:

u_ǫ = v + O(ǫ²), ǫ⁻¹φ_ǫ = −^τ_θ∂_xv + O(ǫ²), ǫ⁻²ψ_ǫ = −_2θ^τ22∂_x²v + O(ǫ²).

All relations are understood with respect to the C T;H^m−1(X) -norm.

Proof: The statement is a straightforward consequence of the previous corollary and the triangle inequality. By adding the same term on both sides of inequality (3.37), we get

ku_ǫ−Pⁿ

j=0

u^(2j)ǫ^2jk+kPⁿ

j=1

u^(2j)ǫ^2jk ≤ k1kFKǫ²ⁿ⁺²+kPⁿ

j=1

u^(2j)ǫ^2jk, where the norms refer toC T;H^m−1(X)

. Since arbitrary realsa, bsatisfy|a−b| ≤

|a|+|b|, the left hand side is shrunk toku_ǫ−u⁽⁰⁾k. The right hand side contains only terms being at least of order ǫ². As the sum extends only over finitely many terms, which are continuous due to the regularity assumptions and thence bounded on the compact time-space domain XT, the right hand side can be estimated for ǫ∈(0,1] by a constant times ǫ². So it is an O(ǫ²)-term.

Repeating the same arguments for the first and second moment, leads to:

kφ_ǫ−ǫ φ⁽¹⁾k ≤ kskFKǫ²ⁿ⁺²+ O(ǫ³) kψǫ−ǫ²ψ⁽²⁾k ≤ ks²− ¹_θkFKǫ²ⁿ⁺²+ O(ǫ⁴)

Division byǫ¹ andǫ² respectively transforms the right hand side into an O(ǫ²)-term

ifn≥1.

Observe that we need at least ˆf_ǫ^[5] to conclude the assertion for φ_ǫ andψ_ǫ.

3.4 Initial conditions and irregular expansions

As the regular expansion considered so far starts with a term of the form u⁽⁰⁾w, it cannot approximate solutions belonging to initial conditions whose leading order does not lie in the span of w ∈ F. This invites the question how arbitrary initial values evolve.

To answer this, we try to find other, non-regular expansions to construct approxi-mate solutions of the lattice-Boltzmann equation¹³. Before we present the working expansion, we discuss an ansatz which turns out to fail.

13For simplicity the source term is set to 0.

We track the presumption that initial values being in span{s,s²−¹_θ}are subject to a fast evolution in comparison with those in span{w}. In this contextfastmeans that they become marginally small after an ǫ-depending time that is short compared with 1 (e.g. ǫ, ǫ², ...).

This motivates the introduction of a scaled time variable t/ǫ which distinguishes the following ansatz

j_ǫ(t/ǫ, x) =j⁽⁰⁾(t/ǫ, x) +ǫj⁽¹⁾(t/ǫ, x) +... (3.39) from the regular expansion. The role ofj_ǫis to approximate the population function f_ǫ just like the comparison function ˆf_ǫ. In order to ensure that (3.39) makes sense we require that the coefficient functions are bounded for ǫ ↓ 0. This entails in particular that ǫ^kj^(k)(t/ǫ, x) = O(ǫ^k). Plugging (3.39) into the lattice-Boltzmann equation, the standard comparison of coefficients¹⁴ yields

ǫ⁻² : 0 = J₀j⁽⁰⁾ ǫ⁻¹ : ∂₁j⁽⁰⁾+S∂_xj⁽⁰⁾ = J₀j⁽¹⁾

ǫ^k−1: ∂₁j^(k)+S∂_xj^(k) = J₀j^(k+1)−chj^(k−1),1iw









(3.40)

where ∂₁ denotes the derivative with respect to the first variable, i.e. t/ǫ(observe:

∂_t =ǫ⁻¹∂₁). The first equation pertaining to the order ǫ⁻² agrees with the corre-sponding result in (3.13) for the regular expansion. So we obtain

j⁽⁰⁾ ∈kerJ₀ ⇒ j⁽⁰⁾(t/ǫ, x) =w⁽⁰⁾(t/ǫ, x)w (3.41) with some unknown functionw⁽⁰⁾. Already now this indicates clearly that (3.39) is not pertinent for our goal because it produces structurally the same term in zeroth order as the regular expansion. Nevertheless let us continue to check whether the ansatz works in principle.

Using (3.41) to compute the mass moment of the second equation in (3.40) h∂₁j⁽⁰⁾,1i + hS∂_xj⁽⁰⁾,1i = hJ₀j⁽¹⁾,1i

∂₁w⁽⁰⁾hw,1i

| {z }

=1

+∂_xw⁽⁰⁾hsw,1i

| {z }

=0

= ω

hj⁽¹⁾,1iw−j⁽¹⁾,1

| {z }

=0

we deduce ∂₁w⁽⁰⁾ = 0, which shows that w⁽⁰⁾ is constant in time. Hence j⁽¹⁾ is solution of the equation

J0j⁽¹⁾=∂xw⁽⁰⁾sw.

According to lemma 3.2 this equation is solvable since hsw,1i = 0. Furthermore this yields

j⁽¹⁾=w⁽¹⁾w−τ ∂xw⁽⁰⁾sw. (3.42) Let us now turn to the zero order equation being the third one in (3.40) withk= 1:

∂1w⁽¹⁾w+∂xw⁽¹⁾sw−τ ∂xw⁽⁰⁾s²w+cw⁽⁰⁾w=J0j⁽²⁾ (3.43)

14Strictly speaking, the comparison of coefficients applies for polynomials and power series if the coefficients are independent of the expansion variable. This situation is not given here. However the boundedness of the coefficients is enough to justify the procedure.

3.4. Initial conditions and irregular expansions 147

Again, this equation is only solvable if the mass moment of the left hand side vanishes which provides the equation to determinew⁽¹⁾ more precisely:

∂₁w⁽¹⁾−^τ_θ∂_xw⁽⁰⁾+cw⁽⁰⁾= 0.

Asw⁽⁰⁾ does not depend on time we conclude w⁽¹⁾(t/ǫ, x) = t

ǫ

τ

θ∂_xw⁽⁰⁾(x)−cw⁽⁰⁾(x)

+ const.

However, this result violates the assumption that the order functions j⁽⁰⁾,j⁽¹⁾, ...

remain bounded for fixed t and x as ǫ goes to 0. To overcome this inconsistency, we are obliged to equate the coefficient behind ^t_ǫ with 0, which implies w⁽¹⁾ to be constant in time as well. Thus we are led to a differential equation determining the spatial dependence ofw⁽⁰⁾:

cw⁽⁰⁾= ^τ_θ∂_xw⁽⁰⁾.

For c6= 0 the solution is given by an exponential function. This, however, cannot comply with the periodic boundary conditions. Therefore we must infer that w⁽⁰⁾ vanishes if c6= 0. In any case, w⁽⁰⁾ must be constant in space and time. So (3.43) becomes

J₀j⁽²⁾ =∂_xw⁽¹⁾sw.

Since this representation is analogous to (3.42) and since the equation for j⁽³⁾ is structurally the same as (3.43) for j⁽²⁾, we see that we can repeat the arguments to infer that w⁽¹⁾ is constant too. This shows finally that only trivial or at most constant solutions of the lattice-Boltzmann equation can be expanded by (3.39).

Due to this reason the ansatz must be rejected.

In order to obtain a more enlightening expansion we have to choose the time scaling t/ǫ^α such that the equation in the lowest order is not of the type J₀x = 0. It is quickly seen that this requirement is fulfilled for α = 2. Therefore, let us consider the expansion

j_ǫ(t/ǫ², x) =j⁽⁰⁾(t/ǫ², x) +ǫj⁽¹⁾(t/ǫ², x) +... , (3.44) resulting in the subsequent equations for the order functions:

ǫ⁻² : ∂₁j⁽⁰⁾ = J₀j⁽⁰⁾ ǫ⁻¹ : ∂1j⁽¹⁾+S∂xj⁽⁰⁾ = J0j⁽¹⁾

ǫ^k−2: ∂1j^(k)+S∂xj^(k−1) = J0j^(k)−chj^(k−2),1iw









(3.45)

In contrast to the order functions of the regular expansion, we obtain forj⁽⁰⁾,j⁽¹⁾, ...

no PDEs but ODEs, since the equations do not involve the spatial derivative of the highest occurring order function. So, thej^(k)’s come out as solutions of ODEs with respect to the scaled time variabler =t/ǫ² having the structure

∂₁j^(k)−J₀j^(k) =−S∂_xj^(k−1)−chj^(k−1),1iw.

The right hand side is a true inhomogeneity as it depends only on j^(k−1) being determined by the previous equation. Thus the solution is formally given by means of the variation of constants formula:

j^(k)(t/ǫ², x) = e^tJ0/ǫ2j^(k)(0, x)

− e^tJ0/ǫ2 Z _t/ǫ²

0

e^−rJ0

S∂xj^(k−1)(r, x) +chj^(k−1)(r, x),1iw dr The leading order function j⁽⁰⁾ satisfies the homogeneous equation∂₁j⁽⁰⁾=J₀j⁽⁰⁾. Let us now consider each of the three basis vectorsw,s,s²−¹θ ∈ F as initial value for j⁽⁰⁾. The following computation shows that these vectors constitute an eigenbasis of the collision operator J₀:

J₀w =ωh

hw,1iw−wi

= 0, J₀s =ωh

hs,1iw−si

=−ωs, J₀(s²−¹_θ) =ωh

hs²− ¹_θ,1iw−s²−¹_θi

=−ω(s²−¹_θ).

Since it is assumed that ω is strictly positive, the eigenvalues of s and s²− ¹_θ are negative which has a strong impact on their dynamics. So the solution associated with s and s²− ¹_θ as initial values decreases exponentially¹⁵ in the fast time t/ǫ². Using the variation of constants formula, it becomes clear that the exponential de-cay devolves upon the subsequent order functions j⁽¹⁾,j⁽²⁾, ... too. A fortiori, they remain bounded for fixedt, xasǫ tends to 0, which justifies the ansatz (3.45).

As far as the first eigenvectorwis concerned, the expansion displays a quite different behavior. Due to J₀w = 0, the initial value j_ǫ(0, x) = w⁽⁰⁾(x)w implies that j⁽⁰⁾ is constant in time, i.e. j⁽⁰⁾(t/ǫ², x) = w⁽⁰⁾(x)w. For j⁽¹⁾ we obtain then the initial value problem

∂1j⁽¹⁾+J0j⁽¹⁾ =∂xw⁽⁰⁾(x)ws with j⁽¹⁾(0, x) = 0.

Since ws is also an eigenvector of J₀ pertaining to the eigenvalue −ω, this yields j⁽¹⁾(t/ǫ², x) =τe^−ωt/ǫ2∂_xw⁽⁰⁾(x)ws−τ ∂_xw⁽⁰⁾(x)ws.

The exponentially decaying part is to adjust the initial condition while the second part (solution of the inhomogeneous equation) is constant in time and resembles the expression for f⁽¹⁾ in equation (3.38). Without going into details of further calcula-tions to compute higher order funccalcula-tions, we may note that the expansion appears also reasonable in the case of a w-proportional initial condition. In opposition to the other cases it contains constant terms. Once the decaying terms have vanished practically, a regular expansion is left¹⁶ whose mass moment is given by the time

15For the initial value j⁽⁰⁾(0, x) = w⁽⁰⁾s we obtain the solution j⁽⁰⁾(t, x) = w⁽⁰⁾(x)e^tJ0/ǫ2s = w⁽⁰⁾(x)e^−ωt/ǫ2s. Ifs²−¹_θ orswis chosen instead ofs, a completely analogous result is found.

16In the short time scale the “diffusive” decay ofw⁽⁰⁾wis not observable.

Im Dokument Analysis of Lattice-Boltzmann Methods : Asymptotic and Numeric Investigation of a Singularly Perturbed System (Seite 157-161)