Two-scale expansion and resolution of the initial layerlayer

Hitherto, our analysis of the singular IBVP (2.29) is based on the classical tool of Fourier series. Even though this technique offers the advantage of providing quasi-explicit solutions, analytic representations may be sometimes rather cumbersome to extract qualitative properties.

In this section we attack the singular IBVP by asymptotic expansions with respect to the perturbation parameter ǫ. As mentioned at the beginning of the previous section, this approach works independently of the possibility to apply Fourier series and is particularly suitable for non-linear problems. In contrast to the Fourier ap-proach, the asymptotic expansion we will choose below has the virtue to offer much insight into the “internal dynamics” of the perturbed equation. The temporal evo-lution of the deviation uǫ−u is in the limelight of discussion to understand the convergence in detail. By means of the asymptotic expansion it is possible to derive equations telling us, where and why the error is generated and by what sources it is fed.

Although the perturbed problem (2.29) could be tackled directly by asymptotic expansions, we prefer to profit from the Fourier expansion in so far as it reduces the PDE problem to a bunch of structurally equal ODE problems. Since the qualitative behavior of the singular limit is fully contained in each of the Fourier coefficient functions, it is enough to concentrate here on a single Fourier mode. Hence we drop from now on the index n. Recall that every Fourier coefficient function of the perturbed problem solves an IVP of the type

EQ : ǫ²τσ¨_ǫ+ ˙σ_ǫ+λσ_ǫ= 0 IC : σǫ(0) =α ∧ σ˙ǫ(0) =β

)

(2.69) with the limit problem

EQ : σ˙ +λσ = 0 IC : σ(0) =α

)

. (2.70)

In order to analyze how the solution of the IVP (2.69) behaves for ǫ ↓ 0, we at-tempt to representσ_ǫby an asymptotic expansion. The simplest ansatz is aregular expansion corresponding to a (truncated) power series inǫ

σ_ǫ(t) =ς⁽⁰⁾(t) +ǫ²ς⁽²⁾(t) +ǫ⁴ς⁽⁴⁾(t) +... (2.71) with the asymptotic order functions ς^(2k), k= 0,1,2, ..., as time depending coeffi-cients. Plugging (2.71) into (2.69) and equating terms of the same order inǫ, yields the subsequent IVPs for the order functions:

ǫ⁰ : ς⁽⁰⁾(0) =α ς˙⁽⁰⁾ +λς⁽⁰⁾ = 0 ǫ² : ς⁽²⁾(0) = 0 ς˙⁽²⁾ +λς⁽²⁾ =−τς¨⁽⁰⁾ ǫ^2k: ς^(2k)(0) = 0 ς˙^(2k)+λς^(2k)=−τς¨^(2k−2)

(2.72)

2.3. Two-scale expansion and resolution of the initial layer 109

Obviously the leading order function ς⁽⁰⁾ satisfies (2.70), that is why we obtain ς⁽⁰⁾ =σ. This confirms the convergence result of proposition 2.3, namely:

σ_ǫ =ς⁽⁰⁾+ǫ²ς⁽²⁾+...=σ+ O(ǫ²). (2.73) Observe, that the second initial condition ˙σǫ(0) = β has been completely ignored to derive (2.72). This condition cannot be taken into account, since otherwise the IVPs in (2.72) would forfeit their well-posedness. This indicates that the regular expansion (2.71) is unable to reflect the full behavior of σ_ǫ.

Motivation & computation of a general two-scale expansion

To get an idea how to improve ansatz (2.71) we take a look at the right plot of figure 2.3, which compares the evolution of ˙σ_ǫ and ˙σ. Especially for small values of ǫ, the curves representing ˙σ_ǫ appear as superpositions of two processes.

Despite of being not compatibly initialized, ˙σ_ǫ tries to attain the initial value of ˙σ.

This process, which dominates the initial phase, corresponds to a quick relaxation, starting from the incompatible initial value 0 and tending the more rapidly towards

˙

σ(0) the smallerǫis chosen. Simultaneously, another process is triggered that seems to counteract the first one. This second process mimics the diffusive decay of the limit function ˙σ. In contrast to the relaxation, the diffusion evolves relatively slowly and becomes clearly visible only after the relaxation has lost its prevalence.

While the curves representing ˙σǫ converge towards ˙σ outside some vicinity of 0, they behave differently near 0. It seems as if they approximate a vertical line that cannot be described as graph of a single-valued function. This means that the limit function becomes discontinuous, jumping instantaneously from the actual incom-patible initial value 0 to ˙σ(0).

This indicates that the relaxation process gets accelerated as ǫ tends to zero. To visualize the process better, ˙σ_ǫis plotted versus the scaled time t/ǫ² in figure 2.5 c) where a usual convergence behavior can be observed. So we are led to the conclusion, that apparently ˙σǫexhibits two time-scales similar to the functiont7→t²+ sin(t/ǫ²) representing a parabola modulated by a sinusoidal oscillation. If we look at the long-term behavior of this function, then we recover the parabola. Zooming into a time interval of lengthǫ≪1, the quadratic growth of the parabola becomes hardly per-ceptible, whereas the oscillations get very striking.

Inspired by these observations we are inclined to assume that ˙σ_ǫ and hence σ_ǫ as well depend on the time in a two-fold manner such that they are described by a two-scale function²⁷. Concretely we suppose that the variablet occurs either alone (slow time) or in combination with the factorǫ⁻², i.e. t/ǫ² (fast time²⁸). Therefore

27Generally, the definition ofmulti-scale dependenceremains somewhat vague, because this no-tion combines some degree of arbitrariness. For instance, the identity funcno-tiont7→tcan be written quite artificially in the formt=¹₅t+⁴₅ǫ²(t/ǫ²), which looks like a veritable multiple-scale function.

28Ifǫis a small parameter less than 1, thent=ǫ² impliest/ǫ²= 1> ǫ². This means thatt/ǫ² grows much faster thant.

0 0.05 0.1 0.15

Figure 2.5: Inspection of the initial layer: Subplot a) shows the long-term behavior (diffusive decay) of the derivative σ˙_ǫ as a function of time for different values of ǫ.

It can be seen clearly, that the curves coincide almost outside the range of the initial layer manifesting the convergence. Subplot b) yields a close-up of the so called initial layer, that represents the short-term relaxation process to the initial value of the limit function σ. In subplot c)˙ σ˙_ǫ is plotted versus the scaled time t/ǫ² being referred to as the fast time. Comparing c) with a), the curves reveal now the converse behavior, because they agree near0and diverge as the rapid time increases. The fat dots indicate the time dilation depending on ǫ. For instance, the time dilation factor is 16 times larger for the smallest value of ǫ(dashed black) than for the second smallest one (solid cyan). Parameter setting: τ = 0.3, λ = 9π²τ, ǫ= 0.15(dashed blue), 0.1(solid red), 0.08(dash-dotted green), 0.06(dashed magenta), 0.04(solid cyan), 0.01(dashed black).

we introduce two new variables: t/ǫ² is replaced by r – the rapid time variable – whereas t alone is substituted bys– the slow time variable. Although this setting decouples both of the time scales in an unnatural²⁹way, asr andsare independent of each other while t/ǫ² and tare not, it is reasonable to do so in order to simplify the analysis of the problem.

σ_ǫ(t) =σ⁽⁰⁾(t/ǫ², t) +ǫ²σ⁽²⁾(t/ǫ², t) +ǫ⁴σ⁽⁴⁾(t/ǫ², t). . . (2.74) As usual, theσ^(2k)’s are referred to asasymptotic order functions. Expansion (2.74) makes sense only if we can apply thecomparison of coefficients³⁰. This requires the σ^(2k)’s to remain bounded if ǫ tends to 0. Therefore we postulate the existence of a constant C >0 such that

∀t∈[0, T], ∀ǫ∈(0,1) : |σ^(2k)(t/ǫ², t)|< C^2k. (2.75) Thiscondition of boundedness³¹ turns out to be crucial, when we extract the equa-tions governing the evolution of the order funcequa-tions.

29From a mathematical point of view this is just a change of coordinates from thet-ǫplane to ther-splane.

30Two (finite or infinite) expansions of the type (2.74) are equal, if and only if all corresponding asymptotic order functions are equal. This can be seen successively by sendingǫto 0. Thanks to thecondition of boundednesswe obtain the equality of the zeroth order. Subtracting this, dividing byǫ² and sendingǫonce more to zero we get the equality of the second order and so forth.

31Observe that this condition is less restrictive than uniform boundedness with respect tok. If

2.3. Two-scale expansion and resolution of the initial layer 111

By computing the first and second derivative of σ^(2k) with respect to the time t

d

we recognize that differentiation will mix the orders of ansatz (2.74): hence the derivative of an order function is not equal to the order function of the derivative

˙

σ_ǫ as in the case of the regular expansion. Of course, this is due to the fact, that the asymptotic order functions depend on ǫ, if they are written as functions of t.

Plugging (2.74) into (2.69), collecting terms of equal order inǫand performing the standard comparison of coefficients we obtain the following equations

ǫ⁻²: ∂_r²σ⁽⁰⁾ + ω∂_rσ⁽⁰⁾ = 0

ǫ⁰ : ∂_r²σ⁽²⁾ + ω∂_rσ⁽²⁾ =−2∂_r∂_sσ⁽⁰⁾−ω∂_sσ⁽⁰⁾−ωλσ⁽⁰⁾

ǫ^2k:∂_r²σ^(2k+2)+ω∂_rσ^(2k+2)=−2∂_r∂_sσ^(2k)−ω∂_rσ^(2k)−ωλσ^(2k)−∂_s²σ^(2k−2)

(2.76)

fork∈N. We interpret each equation as evolution equation for the highest occur-ring asymptotic order function. Then (2.76) constitutes a sequence of equations, which are successively solvable – starting with the lowest order function σ⁽⁰⁾ ap-pearing as source for σ⁽²⁾ and so forth.

The second order ODEs³² are accompanied by initial conditions, that are derived by plugging (2.74) into the initial conditions of (2.69) and performing once more the comparison of coefficients.

σ⁽⁰⁾(0,0) =α , ∂_rσ⁽⁰⁾(0,0) = 0 σ⁽²⁾(0,0) = 0 , ∂rσ⁽²⁾(0,0) = β−∂sσ⁽⁰⁾(0,0) σ^(2k+2)(0,0) = 0 , ∂_rσ^(2k+2)(0,0) = −∂_sσ^(2k)(0,0)

(2.77)

Formally σ^(2k), k ∈ N₀, is calculated by transforming the scalar second-order dif-ferential equation into a first order system, whereDuhamel’s formula (variation of constants) is applied. So we set

z^(2k)(r, s)≡

Then the corresponding evolution system becomes

∂_rz₁^(2k)= z₂^(2k)

the expansion has infinitely many terms then it is convergent for ǫ <1/C, as it is dominated by the geometric series belonging toCǫ <1.

32It is remarkable, that the resulting system (2.76) of differential equations needs not to be con-sidered as a system of PDEs (in the variabler, s), whose solution would be much more complicated.

HereQ^(2k)represents the associated source term occurring in the equations of (2.76).

In matrix notation this system is equivalently expressed by

∂_rz^(2k)=Az^(2k)+Q^{(2k) 0}₁

k∈N₀. (2.79)

Duhamel’s formula reads now

z^(2k)(r, s) = e^rAy^(2k)(s) + Z r

0

Q^(2k)(ϑ, s) e^{(r−ϑ)A 0}₁

dϑ , (2.80)

where the exponential matrix is explicitly given by e^rA=

1 τ −τe^−ωr 0 e^−ωr

.

With y^(2k) we denote the initial condition depending generally on the slow time variable s that plays the subordinate role of a mere parameter in the differential equation (2.79). Let us finally write down (2.76) in the new notation to see, how the sources look like:

∂_sz⁽⁰⁾ − Az⁽⁰⁾ = 0,

∂_sz⁽²⁾ − Az⁽²⁾ =−

2∂_sz₂⁽⁰⁾+ω∂_sz₁⁽⁰⁾+ωλz₁⁽⁰⁾ ₀

1

,

∂sz^(2k+2)−Az^(2k+2)=−

2∂sz₂^(2k)+ω∂sz^(2k)₁ +ωλz₁^(2k)+∂_s²z₁^(2k−2) ₀

1

.

(2.81)

These evolution equations are supplemented by the initial conditions z⁽⁰⁾(0,0) = ^α₀

, z⁽²⁾(0,0) = _β−∂ ⁰

sz⁽⁰⁾₁ (0,0)

and z^(2k+2)(0,0) = _−∂ ⁰

sz^(2k+2)₁ (0,0)

becoming in terms of y⁽⁰⁾,y⁽²⁾ and y^(2k+1) y⁽⁰⁾(0) = ^α₀

, y⁽²⁾(0) = _β ⁰

−y˙⁽⁰⁾₁ (0)

and y^(2k+2)(0) = ⁰

−y˙^(2k)₁ (0)

, (2.82) where dots indicate derivatives with respect to s.

Computation of z⁽⁰⁾(r, s). Since the equation forz⁽⁰⁾is homogeneous (Q⁽⁰⁾ ≡0), we obtain from (2.80)

z⁽⁰⁾(r, s) = e^rAy⁽⁰⁾(s) = y₁⁽⁰⁾(s) +τ y₂⁽⁰⁾(s)−τe^−ωry₂⁽⁰⁾(s) e^−ωry₂⁽⁰⁾(s)

!

. (2.83)

In order to determine y⁽⁰⁾(s) we consider the solution formula for z⁽²⁾: z⁽²⁾(r, s) = e^rAy⁽²⁾(s)−

Z _r

0

h2∂_sz₂⁽⁰⁾+ω∂_sz₁⁽⁰⁾+ωλz₁⁽⁰⁾i

(ϑ, s) e^{(r−ϑ)A 0}₁ dϑ

| {z }

=:J⁽⁰⁾

.

Let us compute the integral J⁽⁰⁾ more explicitly. For this we replace z⁽⁰⁾₁ and z₂⁽⁰⁾ by their representation given in (2.83). FurthermoreJ⁽⁰⁾ is decomposed into a sum

2.3. Two-scale expansion and resolution of the initial layer 113

of four integrals, whose specific dependence on the rapid timer is indicated at the right.

J⁽⁰⁾= ^τ₀ Z ^r

0

hωy˙⁽⁰⁾₁ + ˙y₂⁽⁰⁾+ωλy₁⁽⁰⁾+λy₂⁽⁰⁾i

dϑ | ∝r

+ ^τ₀Z r 0

hy˙⁽⁰⁾₂ −λy⁽⁰⁾₂ i

e^−ωϑdϑ | ∝τ(1−e^−ωr) + ⁻₁^τ Z ^r

0

h

ωy˙⁽⁰⁾₁ + ˙y₂⁽⁰⁾+ωλy₁⁽⁰⁾+λy⁽⁰⁾₂ i

e^ω(ϑ−r)dϑ | ∝τ(1−e^−ωr) + ⁻₁^τ Z ^r

0

hy˙₂⁽⁰⁾−λy₂⁽⁰⁾i

e^−ωrdϑ | ∝re^−ωr

Sincey⁽⁰⁾ only depends on the slow times, the rectangular braces are independent of the integration variable ϑ. In particular, the integrands of the first and fourth integral are independent ofϑ.

In the case of the first integral, the integration yields a term of the formrf(s) withf being thes-dependent integrand ωy˙⁽⁰⁾₁ + ˙y₂⁽⁰⁾+ωλy₁⁽⁰⁾+λy₂⁽⁰⁾. After resubstitution of the time, r → t/ǫ² and s→ t, we get tf(t)/ǫ². For f 6≡0 this term can not be uniformly bounded with respect to ǫ ∈ (0,1) if t > 0. Since this term occurs in the first component ofz⁽²⁾, being equal toσ⁽²⁾, the asymptotic order function σ⁽²⁾ violates the condition of boundedness (2.75) if f 6≡0. Therefore, to comply with (2.75), we must require f to vanish. This yields one of the desired equations for determining the components of y⁽⁰⁾

ωy˙₁⁽⁰⁾+ ˙y⁽⁰⁾₂ +ωλy₁⁽⁰⁾+λy⁽⁰⁾₂ = 0. (2.84) Unfortunately, the ansatz does not supply the second equation to determine y⁽⁰⁾₁ and y₂⁽⁰⁾ in a unique way. In fact, we should not expect, that these variables come out univocally, as we have replaced one unknownσ⁽⁰⁾ by two unknownsy⁽⁰⁾₁ , y₂⁽⁰⁾. Regarding the second integral proportional to 1−e^−ωr, it is clear, that the ar-gument from above does not work to produce another equation. In this respect, the fourth integral looks more promising. Integration produces a term of the type re^−ωrf(s), wheref stands now for ˙y₂⁽⁰⁾−λy⁽⁰⁾₂ . After resubstitution of the time we gett/ǫ²e^−ωt/ǫ2f(t). However, due to the exponential function this term is bounded forǫ∈(0,1) andt >0, althoughǫ⁻²occurs as an delusive external factor. So, none of the other integrals gives rise to a second equation.

As we dispose only of one equation for two unknownsy₁⁽⁰⁾, y₂⁽⁰⁾, we can define one of them arbitrarily (in accordance with initial conditions and regularity requirements) and compute the other by equation (2.84). In particular we can choose y₂⁽⁰⁾ to satisfy

˙

y⁽⁰⁾₂ −λy⁽⁰⁾₂ = 0. (2.85)

Observe that this choice is a priori not distinguished, except that it makes the integral J⁽⁰⁾ vanish, what simplifies the further computations.

So we end up with the following IVPs for y⁽⁰⁾₂ andy⁽⁰⁾₁ :

˙

y₂⁽⁰⁾−λy⁽⁰⁾₂ = 0 y₂⁽⁰⁾(0) = 0

)

⇒ y₂⁽⁰⁾≡0 (2.86)

˙

y₁⁽⁰⁾+λy⁽⁰⁾₁ = −τy˙⁽⁰⁾₂ −τ λy⁽⁰⁾₂ = 0 y₁⁽⁰⁾(0) = α

)

⇒ y1(s) =αe^−λs (2.87)

The initial conditions have been read from equation (2.82).

Remark 2.9. We emphasize that the additional imposition of equation (2.85) en-tailing y₂⁽⁰⁾ ≡0 is not compelling. Therefore let us suppose y₂⁽⁰⁾ 6≡ 0. Due to the initial condition y⁽⁰⁾₂ (0) = 0, the componenty⁽⁰⁾₂ (s) can be written as s˜y⁽⁰⁾₂ (s) with some ˜y⁽⁰⁾₂ (s). Using (2.83) we obtain

σ⁽⁰⁾(t/ǫ², t) =y⁽⁰⁾₁ (t) +τ y⁽⁰⁾₂ (t)

| {z }

=O(1)

− ǫ²τe^−ωt/ǫ2t/ǫ²y˜₂⁽⁰⁾(t)

| {z }

=O(1)

| {z }

O(ǫ²)

.

Thus, σ⁽⁰⁾ is not only composed of O(1)-terms but also contains an O(ǫ²)-term, that is expected to appear in the second order function σ⁽²⁾. Even though this is not forbidden, as parts of higher order functions may be incorporated into lower order functions, it is against the idea of the expansion. This justifies our choice as particularly reasonable, since it disentangles the zeroth and second order.

It might be interesting to compute, how the appearance of the O(ǫ²)-term in the zeroth order function σ⁽⁰⁾ affects the second order function σ⁽²⁾. Either this term belongs really to the expansion of σǫ or it comes into play artificially and appears with inverse sign in σ⁽²⁾.

Computation of z⁽²⁾(r, s). The computation proceeds analogously to the previ-ous one, observing that the source Q⁽²⁾ does vanisha posteriori thanks to equation (2.86) and (2.87). So we obtain from (2.80)

z⁽²⁾(r, s) = e^rAy⁽²⁾(s) = y₁⁽²⁾(s) +τ y₂⁽²⁾(s)−τe^−ωry₂⁽²⁾(s) e^−ωry⁽²⁾₂ (s)

!

. (2.88)

In order to determiney⁽²⁾we must appeal to the representation ofz⁽⁴⁾by Duhamel’s formula. Inserting (2.88) into

z⁽⁴⁾(r, s) = e^rAy⁽⁴⁾(s)− Z r

0

h

2∂sz₂⁽²⁾+ω∂sz₁⁽²⁾+ωλz₁⁽²⁾+∂_s²z₁⁽⁰⁾i

(ϑ, s) e^{(r−ϑ)A 0}₁ dϑ

| {z }

=:J⁽²⁾

2.3. Two-scale expansion and resolution of the initial layer 115

yields for the integral J⁽²⁾ J⁽²⁾= ^τ₀ Z ^r

With the same arguments as in the computation ofz⁽⁰⁾ we infer, that the integrand of the first integral must vanish. This results into equation (2.90) for y₁⁽²⁾, that presupposesy₂⁽²⁾ as given. Like (2.86), equation (2.89) is due to a special selection ofy⁽²⁾₂ , such that the second and fourth integral on the right and henceJ⁽²⁾become zero as well.

Computation of z^(2k)(r, s), k ≥ 2. In principle, the computation is repeatable in every order. Assuming that y^(2k−2) has been defined in such a way, that Q^(2k) vanishes, equation (2.80) yields

z^(2k)(r, s) = e^rAy^(2k)(s) = y₁^(2k)(s) +τ y₂^(2k)(s)−τe^−ωry₂^(2k)(s) e^−ωry^(2k)₂ (s)

!

. (2.91) Plugging this into the representation ofz^(2k+2)given once more by (2.80) we obtain

z^(2k+2)(r, s) = e^rAy^(2k+2)(s)

Evidently, the integral J^(2k) has the same structure as J⁽⁰⁾ and J⁽²⁾. This allows us to use the reasoning from above to extract the subsequent equations.

˙

It should be reemphasized, that the equation for y₂^(2k+2) is not enforced by the ansatz. Notice also, that from the fourth order on, the preceding order function appears as source term in the equations of both components. Since (2.93) and (2.92) imply that the integral J^(2k) becomes zero, the condition Q^(2k+2) = 0 is assured, which is necessary to go on likewise with the next order.

Interpretation of the two-scale expansion

In order to construe the equations obtained by the asymptotic expansion, it is advantageous to introduce new variables being more natural and thus simplifying the hierarchic evolution equations derived for they^(2k)s. For k∈N₀ we set:

ζ^(2k):=y₁^(2k)+τ y₂^(2k), φ^(2k):=−τ y^(2k)₂ . According to (2.78) and (2.91) this yields

σ^(2k)(r, s) =z^(2k)₁ (r, s) =y^(2k)₁ (s) +τ y^(2k)₂ (s)−τe^−ωry^(2k)₂ (s)

=ζ^(2k)(s) + e^−ωrφ^(2k)(s). After resubstitution of the time twe get finally

σ^(2k)(t/ǫ², t) =ζ^(2k)(t)

| {z }

regular

+ e^−ωt/ǫ2φ^(2k)(t)

| {z }

irregular

.

This representation shows that each asymptotic order function can be written as the sum of a regular and an irregular part. The regular part depends only on the slow time s=t like the order functions of the regular expansion. In contrast, the irregular part depends on both time scales; furthermore it reveals a multiplicative structure, where the dependence on the rapid time scale r = t/ǫ² is given explic-itly by the exponential function. Even if the second factor φ^(2k) is of exponential growth (cf. evolution equation for φ⁽²⁾ in (2.95)), for ǫsufficiently small the expo-nential function effectuates a decay of the irregular part, that becomes the faster the smaller ǫis chosen. Due to this circumstance the irregular part is called initial layer³³, because it is only “visible” as long as t is of the size ǫ². Therefore it is almost irrelevant for the behavior of σ^(2k) over a time interval of magnitude 1.

Let us now express the IVPs determining the y^(2k)s in terms of the regular part ζ^(2k) and the initial layer function φ^(2k). For φ⁽⁰⁾ and ζ⁽⁰⁾ we obtain from (2.86) and (2.87)

φ⁽⁰⁾ ≡0

(ζ˙⁽⁰⁾+λζ⁽⁰⁾ = 0

ζ⁽⁰⁾(0) =α . (2.94)

33In order to get convergence of the irregular part (initial layer) e^−ωt/ǫ2φ^(2k)(t) to a regular function for ǫ→0, we must write it as a function of the rapid time. Replacing tby ǫ²r yields e^−ωrφ^(2k)(ǫ²r)−−−→^ǫ→0 e^−ωrφ^(2k)(0), so that the modulating functionφis reduced to its initial value.

Ifσǫ(t) is plotted versus the rapid time (cf. figure 2.5), the curves nearly coincide with the graph ofr7→e^−ωrφ^(2k)(0). Therefore it is reasonable to refer to e^−ωras the initial layer (profile).

2.3. Two-scale expansion and resolution of the initial layer 117

Similarly, we obtain from (2.89) and (2.90) (φ˙⁽²⁾−λφ⁽²⁾= 0

φ⁽²⁾(0) =τζ˙⁽⁰⁾(0)−τ β

(ζ˙⁽²⁾+λζ⁽²⁾ =−τζ¨⁽⁰⁾

ζ⁽²⁾(0) =−φ⁽²⁾(0) . (2.95) Eventually, the general case (2.92) and (2.93) leads to:

(φ˙^(2k)−λφ^(2k) =τφ¨^(2k−2)

φ^(2k)(0) =τφ˙^(2k−2)(0) +τζ˙^(2k−2)(0)

(ζ˙^(2k)+λζ^(2k)=−τζ¨^(2k−2)

ζ^(2k)(0) =−φ^(2k)(0) (2.96) These IVPs can be solved³⁴subsequently as well, computing first ζ⁽⁰⁾, then φ⁽²⁾ beforeζ⁽²⁾, thereafter φ⁽⁴⁾ and ζ⁽⁴⁾ and so on.

Convergence of σ_ǫ. In contrast to the higher order-functions, σ⁽⁰⁾ consists only of a regular part, which is the solution of the unperturbed IVP (2.70). Of course, this impliesζ⁽⁰⁾=σ. So we end up with the expansion

σǫ(t) = ζ⁽⁰⁾(t)

| {z }

=σ(t)

+ǫ²

ζ⁽²⁾(t) + e^−ωt/ǫ2φ⁽²⁾(t)

+ ... . (2.97) Similarly to the result obtained by the regular expansion (cf. equation (2.73)), this affirms the assertion of proposition 2.3. It becomes immediately clear, whyσǫshould approximate σ with a deviation of second order:

σ_ǫ(t)−σ(t) =ǫ²

ζ⁽²⁾(t) + e^−ωt/ǫ2φ⁽²⁾(t)

+ ... ⇒ |σ_ǫ(t)−σ(t)|= O(ǫ²). Convergence of σ˙_ǫ. Differentiating (2.97) with respect to the time yields

˙

σ_ǫ(t) = ˙ζ⁽⁰⁾ −ωe^−ωt/ǫ2φ⁽²⁾ +ǫ²

ζ˙⁽²⁾ + e^−ωt/ǫ2φ˙⁽²⁾ −ωe^−ωt/ǫ2φ⁽⁴⁾

+... . (2.98) Differentiation of the initial layer seems to add a term to the zeroth order. However, as the exponential functionǫ7→e^−ωt/ǫ2 fades down much faster than any power of ǫ for t > 0, this term can actually be counted to the second order. Similarly to lemma 2.7 one can prove³⁵

e^−ωt/ǫ2 ≤ ^τ_tǫ² .

Therefore, ˙σ_ǫapproximates ˙σwith second order accuracy, as claimed by proposition 2.4, if tstays away from a neighborhood of 0. Hence we get

|σ˙_ǫ(t)−σ(t)˙ |= O(ǫ²) uniformly for allt > θ,

where θ is some fixed positive time. A special case happens, if φ⁽²⁾ vanishes iden-tically. Because of the absence of the initial layer in the second order, uniform approximation occurs then on the entire non-negative time-ray. This exceptional case is given if and only if φ⁽²⁾ is initialized by 0 or equivalently τζ˙⁽⁰⁾(0)−τ β = 0 which is satisfied for ˙σ_ǫ(0) =β =^! −αλ= ˙σ(0). So the initial layer (in the second order) is suppressed if σ_ǫ is initialized compatibly.

34As far as the successive solution of these IVPs is concerned, we need not care about questions of regularity. Since the starting equations for ζ⁽⁰⁾ andφ⁽²⁾ are homogeneous, their solutions are C^∞and this entails the same smoothness for all the other functions pertaining to higher orders.

35Observe: e^−ωt/ǫ2< M ǫ² ⇔ ǫ⁻²e^−ωt/ǫ2 =:f(ǫ²)≤M. The constantM must be less or equal than the maximum of the functionf(ǫ²).

Further comments on the structure of the deviation σ_ǫ−σ. Now we turn to the regular part of the leading error term σ⁽²⁾. The linearity of the problem permits us once more to decompose easily ζ⁽²⁾ into two parts, that are of different provenance

ζ⁽²⁾ = ζ_N⁽²⁾

| {z }

natural deviation

+ ζ_I⁽²⁾

|{z}

“interaction”

with initial layer

.

ζ_N⁽²⁾ and ζ_I⁽²⁾ are defined by the following IVPs:

( ζ˙_N⁽²⁾+λζ_N⁽²⁾ =−τζ¨⁽⁰⁾ ζ_N⁽²⁾(0) = 0

( ζ˙_I⁽²⁾+λζ_I⁽²⁾= 0

ζ_I⁽²⁾(0) =−φ⁽²⁾(0)

Sinceζ_N⁽²⁾is driven by the solution of the limit problem, it represents the constituent of the deviation, that is “naturally” expected by writing the perturbed equation (2.69) in the form of the limit equation plus a small source, viz. ˙σ_ǫ+λσ_ǫ=−ǫ²τσ¨_ǫ. As this kind of error is encountered in many approximation processes, it is not specific for the singular nature of (2.69).

Like the initial layer,ζ_I⁽²⁾is generated by incompatible initializationβ6=−αλ. How-ever, it reveals the same decay rate as the solution of the limit problem ζ⁽⁰⁾. This shows that an incompatible initialization does not only spark the rapidly vanishing initial layer but also affects the long term behavior of σ_ǫ.

Just as the deviation terms in the second order are driven by the exact solution and the initial incompatibility, they feed terms of fourth order, that appear again as initial conditions and sources for the sixth order and so forth. Asymptotically, the prevailing terms are, of course, those of second order in ǫ.

Observe that ζ_N⁽²⁾ is equal to ς⁽²⁾ of the regular expansion (2.71), as both satisfy the same IVP. Further terms in the second order of the two-scale expansion (2.97) are masked out by the regular expansion in (2.71). This shows clearly that a full analysis of the singular limit problem requires a two-scale expansion. Nevertheless, the main feature, namely the convergence ofσ_ǫ toσ, is correctly predicted even by the regular expansion.

It should be pointed out, that the different constituents of the error (initial layer, natural part, interaction), that we have identified, can be found less precisely in the estimates performed to prove proposition 2.3 and 2.4.

Smooth initialization up to arbitrary order. The compatible initialization

˙

σǫ(0) =−αλ suppresses only the initial layerφ⁽²⁾ in second order, while the initial layers in higher orders, i.e. φ⁽⁴⁾, φ⁽⁶⁾, ...persist in general. These become noticeable, if we consider the convergence of higher derivatives ¨σ_ǫ → σ,¨ ...

σ_ǫ → ...

σ , .... If we initialize ˙σǫ by a truncated expansion

˙

σ_ǫ(0) =β⁽⁰⁾+ǫ²β⁽²⁾+ǫ⁴β⁽⁴⁾+... , (2.99) it is possible to “switch off” arbitrarily many initial layers by a suitable choice of the coefficients β^(2k). In the sequel we determine the initial condition which removes the initial layers up to order 2n+ 2. We call such an initialization smooth up to

2.3. Two-scale expansion and resolution of the initial layer 119

order 2n+ 2. This ensures that the leading derivatives of σ_ǫ converge uniformly in a neighborhood of 0 and hence on [0,∞).

Inserting (2.97) into (2.99), we obtain by equating terms of same order in ǫ (com-parison of coefficients):

ǫ⁰ : φ⁽²⁾(0) =τ

ζ˙⁽⁰⁾(0)−β⁽⁰⁾ ǫ^2k: φ^(2k+2)(0) =τ

ζ˙^(2k)(0) + ˙φ^(2k)(0)−β^(2k) (2.100) Thus we discover, how the coefficients β^(2k) should be adjusted. First, the known

ζ˙^(2k)(0) + ˙φ^(2k)(0)−β^(2k) (2.100) Thus we discover, how the coefficients β^(2k) should be adjusted. First, the known

Im Dokument Analysis of Lattice-Boltzmann Methods : Asymptotic and Numeric Investigation of a Singularly Perturbed System (Seite 120-139)