A Comparison of Approximate Models for

Radiation in Gas Turbines

M. Frank, M. Seaıd, J. Janicka, A. Klar,R. Pinnau and G. Thommes

Fachbereich Mathematik

Technische Universitat Darmstadt

D–64289 Darmstadt, Germany

andFachbereich Maschinenbau

Technische Universitat Darmstadt

D–64289 Darmstadt, Germany

Abstract

Approximate equations for radiative heat transfer equations coupled to anequation for the temperature are stated and a comparative numerical studyof the different approximations is given. The approximation methods con-sidered here range from moment methods to simplified PN–approximations.Numerical experiments and comparisons in different space dimensions andfor various physical situations are presented.

Key words. Radiative Transfer, SPN–Approximations, Moment Methods, Fre-quency averages, Discontinuous coefficients

AMS(MOS) subject classification.

1 Introduction

During the last years the interest in numerically tractable approximations to theradiative heat transfer equations drastically increased, for example because of thefact that simulation tools for modern gas turbine combustion chambers need topredict accurately the load of the walls due to thermal radiation.

In this paper we consider the radiative transfer equations stated on a boundeddomain D ⊂ R3 in the form

ε2∂tT + ε2u · ∇T = ε2∇ · (k0∇T ) +

∫ ∞ν0

∫S2

κ(I −B)dΩ dν (1.1a)

∀ν > ν0, Ω ∈ S2 : εΩ · ∇I + (σ + κ)I =σ

4π

∫S2

IdΩ + κB (1.1b)

1

supplemented with boundary data

εk n · ∇T = h(Tb − T ) + απ

∫ ν0

0

B(Tb, ν)−B(T, ν) dν (1.1c)

for n · Ω < 0 : I(Ω) = ρI(Ω′) + (1− ρ)B(Tb(x)) (1.1d)

In these equations, I(x,Ω, ν, t) denotes the specific radiation intensity for fre-quency ν at point x ∈ D travelling in direction Ω ∈ S2 at time t ≥ 0. B denotesPlanck’s function

B(ν, T ) =2hPν

3

c2

(ehP ν

kBT − 1)−1

for black body radiation, which involves Planck’s constant hP , Boltzmann’s con-stant kB and the speed of light in vacuum c. T (x, t) denotes the material tem-perature and Tb is the exterior temperature on the boundary. ν0 denotes theboundary of the opaque part of the spectrum. The equations contain the absorp-tion and scattering parameters κ(ν, T ) and σ(ν, T ). k0 is the heat conductioncoefficient. The outside radiation is assumed to be known for the ingoing direc-tions (i.e. n · Ω < 0) on the boundary, where we denote the outward normal on∂D by n. The reflected angle is Ω′ = Ω − 2(n · Ω)n. ρ = ρ(Ω) denotes the partof the radiation which is reflected. Finally, ε denotes a dimensionless parameterdescribing the distance from an optically thick medium.

ε =1

xrefκref,

where xref and κref are reference length scale and reference absorption rate re-spectively.

Hierarchies of models approximating (1.1) have been derived by various authors,[14, 9, 11, 10, 7] and many others. They range from moment methods closed byentropy principles to diffusive methods like the SPN–equations. Clearly, thesemodels will differ in their accuracy and numerical complexity. They are usuallyfaster than direct solution methods for (1.1), but less accurate.

2 Approximate equations

In this section we briefly describe different approximate models for the RHTequations.

2.1 Diffusive Approximations

The following approximations are obtained by an asymptotic procedure, see [11].

2

2.1.1 P1–Approximation

The first approximation yields a well known model approximating the transportequation (1.1) to O(ε2). The classical P1 equation reads

∂tT + u · ∇T = ∇ · (k0∇T ) +

∫ ∞ν0

∇ ·(

1

3(σ + κ)∇Φ

)dν, (2.1a)

∀ν ≥ ν0 : −ε2∇ ·(

1

3(σ + κ)∇Φ

)+ κΦ = κ(4πB), (2.1b)

where Φ ∼∫S2 IdΩ is the total incident radiation. System (2.1) is supplemented

with Robin–type boundary data:

∀ν ≥ ν0 : Φ(x) +

(1 + 3r2

1− 2r1

2ε

3(κ+ σ)

)n · ∇Φ(x) = 4πBb(x), (2.2)

where Bb(x) = B(ν, Tb(x)). The boundary condition for the temperature is thesame as before. ri are values depending on the reflectivity ρ, see [11].

2.1.2 SP2–Approximation

The second approximation to O(ε4) is

∂T

∂t+ u · ∇T = ∇ · k0∇T +

∫ ∞ν0

∇ · 1

3(σ + κ)∇ξ dν, (2.3a)

∀ν ≥ ν0 : −ε2∇ · 3

5(σ + κ)∇ξ + κξ = κ(4πB), (2.3b)

where

ξ = Φ +4

5(Φ− 4πB). (2.4)

The boundary conditions are

∀ν ≥ ν0 : ξ(x) +

(1 + 3r2

1− 4r3

4ε

5(κ+ σ)

)n · ∇ξ(x)

= 4πB(x) +

(1− 2r1

1− 4r3

6

5

)[4πBb(x)− 4πB(x)

]. (2.5)

3

2.1.3 SP3–Approximation

The third approximation is the SP3 equation:

∂tT + u · ∇T = ∇ · (k0∇T ) +

∫ ∞ν0

∇ ·(

1

(σ + κ)∇(a1Ψ1 + a2Ψ2)

)dν, (2.6a)

∀ν ≥ ν0 : −ε2∇ ·(

µ21

(σ + κ)∇Ψ1

)+ κΨ1 = κ(4πB), (2.6b)

∀ν ≥ ν0 : −ε2∇ ·(

µ22

(σ + κ)∇Ψ2

)+ κΨ2 = κ(4πB). (2.6c)

Φ is found as linear combination of Ψ1 and Ψ2. System (2.6) is supplementedwith Robin–type boundary conditions.

α1ψ1(x) +ε

κn · ∇ψ1(x) = −β2ψ2(x) + η1Bb, (2.7a)

α2ψ2(x) +ε

κn · ∇ψ2(x) = −β1ψ1(x) + η2Bb. (2.7b)

µi, ai, αi, βi, ηi are appropriate constants, see again [11]. These equations approx-imate the full RHT equations with order O(ε6). For details see [9, 11]. We notethat – based on the SPN equations – one can derive frequency averaged equations.We refer to [10] for an example.

2.2 Entropy-Moment Approximations

These approximations are based on moment methods where the closure relationis obtained by entropy maximization.

2.2.1 Full Space Entropy-Moment (EM) Approximation

The Levermore entropy approach [14, 5] for radiation yields the following systemfor the frequency independent case κ = constant and ν0 = 0 :

ε2∂tT + ε2u · ∇T = ε2∇ · (k0∇T ) + κ(Φ− 4πB(T )) (2.8a)

ε∇ · F + κΦ = 4πB(T ), (2.8b)

ε∇ · (D(f)Φ) + (κ+ σ)F = 0. (2.8c)

HereB(T ) denotes the frequency averaged Planck functionB(T ) =∫∞

0B(T, ν)dν =

aT 4. The relative flux is given by f = F/Φ and the Eddington factor is

D(f) = D =1− χ

2I +

3χ− 1

2n⊗ n, n =

f

‖f‖, χ =

3 + 4 ‖f‖2

5 + 2√

4− 3 ‖f‖2.

4

Here, Φ approximates the frequency averaged incident radiation∫∞

0

∫S2 I dΩdν

and F is an approximation for∫∞

0

∫S2 Ω I dΩdν. A model including frequency

dependence can be found in [17]. Related models can be found in [16, 12, 2, 15, 1].A more general approach is described in the next subsection.

2.2.2 Partial Space Entropy-Moment (PSEM) Approximation

For a function g = g(ν,Ω), where ν denotes frequency and Ω denotes direction,we define

〈g〉A =

∫A

∫ ∞ν0

g(ν,Ω)dνdΩ.

Here, A denotes the set of the angular integration. For example for a half spacemoment model we have A ∈

S2

+,S2−

. Here, S2+ = Ω ∈ S2 : Ωx > 0 is the

positive half sphere and S2− is defined analogously. We define the corresponding

moments by

ΦA = 〈I〉AFA = 〈ΩI〉APA = 〈(Ω⊗ Ω)I〉A.

where I = I(ΦA, FA) is an approximation of I found by entropy minimization.Let A be a partition of the unit sphere S2. Then the moment system reads forthe case of constant κ:

ε2∂tT + ε2u · ∇T = ε2∇ · (k0∇T ) + κ(∑A′∈A

ΦA′− < B(T ) >S2)

ε∇FA + (κ+ σ)ΦA =σ

4π〈1〉A

∑A′∈A

ΦA′ + κ〈B(T )〉A

ε∇PA + (κ+ σ)FA =σ

4π〈Ω〉A

∑A′∈A

ΦA′ + κ〈ΩB(T )〉A

for all A ∈ A. Note, that the above is a hyperbolic system with relaxationterms as before. For a derivation in the one-dimensional case we refer to [7].Further investigations can be found in [6, 8]. Different approaches to partialspace methods can be found for example in [4]. In the computations below weuse a quarter space method in 2D, dividing the angular space into 4 quadrants.Models including frequency dependence can be developed along the same lines asin the full moment case.

5

3 Numerical Comparisons for fixed temperature

3.1 One-dimensional situation

We specify a test configuration to compare the different models. First, we considera fixed temperature profile in 1D, where the temperature increases linearly from1000 K to 1800 K. The 1-band case is assumed with σ and κ constant. Thus, weconsider in D = [0, 1] the following equations:

µ∂I

∂x+ (σ + κ)I =

σ

2

∫ 1

−1

I(x, µ′)dµ′ + κB(T ((x)),

whereB(T ) = aT 4, T (x) = 1000 + 800x, Tb(x) = T (x), ρ = 0.

We compare in Figure 5.1-5.2 the different approximate equations for the casesσ = 1, κ = 1 and σ = 0.1, κ = 0.01 with the transport solution, i.e. the solutionof the RHT equations. In the case considered here, especially the half momentmethod gives very accurate results.

A comparison of the CPU times for the solution of the full radiative transfer equa-tion and for the solution of the approximate equations depends on the methodused for the elliptic equations and on the number of angular discretizations forthe full radiation problem. To solve the radiative transfer equations we have useda multi level method, compare [3]. For the solution of the diffusive equations weuse a standard method. A kinetic scheme combined with a Newton iteration isused for the half space moment method. Roughly the solution of the SP3 equa-tions takes about twice as much time as the solution of the P1 equations. Thesolution of the half space moment model takes about 10 times as much time andthe solution of the full transport problem is between 2 and 3 orders of magnitudeslower as the P1 solution.

3.2 Two-dimensional situations

3.2.1 Constant coefficients

In the second numerical test we fix the temperature profile in the 2D unit square.We consider D = [0, 1]× [0, 1] and the equations

µ∂I

∂x+ η

∂I

∂y+ (σ + κ)I =

σ

4π

∫ π

0

∫ 2π

0

I(x, y, µ′, η′)sinη′dµ′dη′ + κB(T (x, y)),

where B(T ) = aT 4, T (x, y) = 1000 + 400(x+ y), Tb(x, y) = T (x, y), ρ = 0.

6

The comparison of the direct solution with the results of the different approximatemodels for the case σ = 1, κ = 1 can be found in Figure 5.3. The case σ = 0.1,κ = 0.01 along the diagonal x = y and along y = 0.5 is shown in Figure 5.4 andFigure 5.5, respectively.

For these computations we have used for the partial space moment approximationa quarter space method and again a kinetic scheme combined with a Newton iter-ation to solve the resulting nonlinear system of equations. The diffusive equationsare solved using standard methods.

3.2.2 Discontinuous coefficients

The second two-dimensional testcase is a situation with spatially dependent dis-continuous absorption coefficents κ. We consider a problem with D = [0, 1] ×[0, 10],D0 = [0.45, 0.55]× [4.5, 5.5]. B(T ) = aT 4 with T (x, y) = 1000K in D0 andT (x, y) = 1800K in D \ D0. Tb(x, y) = T (x, y), ρ = 0. κ = κ0 in D0 and κ = κ1

in D \ D0.

Again the solutions of the different approximate equations are compared with thedirect solution of the RHT equations. We choose σ = 1 and consider the casesκ0 = 3, κ1 = 1 and κ0 = 1, κ1 = 0.1, see Figure 5.6 and 5.7.

4 Numerical Comparisons for the full heat trans-

fer equations

In this section we consider the radiative transfer equations as above. However,the temperature is here not fixed, but determined by the heat transfer equationdescribed in the beginning.

4.1 One-dimensional situation

As before we consider as a first step a 1D situation, D = [0, 1]. The equations are

ε2∂tT + ε2u∂xT = ε2k0∂xxT + 2π

∫ ∞0

∫ 1

−1

κ(I −B)dµ dν

εµ∂xI + (σ + κ)I =σ

2

∫ 1

−1

I(x, µ′)dµ′ + κB(T ((x)).

B(T ) = aT 4.

We consider σ = 0, u = 0, k0 = 1 and ε = 1. κ is chosen equal to 1 and 0.1,respectively. Here we choose Dirichlet boundary conditions: I(µ) = B(Tb), µ > 0

7

and T = Tb with Tb = 1000K at the left boundary x = 0. I(µ) = B(Tb), µ < 0and T = Tb with Tb = 1800K on the right boundary x = 1.

A comparison of the solutions of the different approximate equations with theRHT equations can be found in Figure 5.8 - Figure 5.9 for the cases κ = 1 andκ = 0.1.

A comparison of the CPU times for the solution of the full radiative transfer equa-tion and for the solution of the approximate equations depends on the methodused for the elliptic equations and on the number of angular discretizations forthe full radiation problem and gives results as in the previous section.

4.2 Two-dimensional situations

In the second numerical test the 2D unit square D = [0, 1]× [0, 1] is considered.The equations are

ε2∂tT +ε2u ·∇T = ε2∇·(k0∇T )+(

∫ ∞ν0

∫ π

0

∫ 2π

0

I(x, y, µ′, η′)sinη′dµ′dη′−4πB)dν

µ∂I

∂x+ η

∂I

∂y+ (σ + κ)I =

σ

4π

∫ π

0

∫ 2π

0

I(x, y, µ′, η′)sinη′dµ′dη′ + κB(T (x, y)).

Boundary conditions are Dirichlet as before. The boundary temperature Tb ischosen as a linearly increasing function from 1000 K to 1400 K on the boundariesx = 0 and y = 0 of the square and from 1400 K to 1800 K for the boundariesx = 1 and y = 1.

The results along y = 0.5 for the different approximate models can be found inFigure 5.10-5.11. We consider again an optically thicker and an optically thinnercase: κ = 1 and κ = 0.1. σ is chosen equal to 0 for both computations. Theother parameters are chosen as in the 1D case.

4.3 Coupling with Navier Stokes

Finally, we also consider the full moment model. In the last test we couple thissystem to the Navier–Stokes equation and simulated a flame–like flow in 3d. Thecomputed temperature can be found in Figure 5.12. The solution is comparedwith simulations relying on the P1–equations (2.1). While there is no visibledifference in the computed temperatures, we depict in Figure 5.13 the norm of therelative flux |f |. Note, that the P1 solution gives physically unreasonable results,i.e. |f | > 1. This is well known, see for example [13] and true for all diffusiveapproximations. In contrast the moment method gives a physically reasonableresult. The bound |f | < 1 is fullfilled for all moment methods constructed here.

8

5 Conclusions

All approximate models give reasonable results in situation which are sufficientlydiffusive. Partial space moment models give accurate approximations even inoptically thinner regimes. The numerical effort for all approximations is in generalsmaller than the time needed for the solution of the full RHT even if a fast iterativemethod is used for the later equations. A comparison and combination of theseapproximate methods with efficient multigrid methods for the full radiative heattransfer equations is currently under investigation.

Acknowledgements

The authors acknowledge support from the German Research Foundation (DFG),SFB568.

References

[1] Brunner T. A. and J.P. Holloway. One-dimensional Riemann solvers and themaximum entropy closure. J. Quant. Spect. Rad. Transf., 69:543–566, 2001.

[2] A.M. Anile, S. Pennisi, and M. Sammartino. A thermodynamical approachto Eddington factors. J. Math. Phys., 32:544–550, 1991.

[3] C.T. Banoczi, J.M.and Kelley. A fast multilevel algorithm for the solutionof nonlinear systems of conductive-radiative heat transfer equations. SIAMJ. Sci. Comput. 19, 19:266–279, 1998.

[4] P.J. Clause and M. Mareschal. Heat transfer in a gas between parallel plates:Moment methods and molecular dynamics. Physical Review A, 38:4241–4252, 1988.

[5] B. Dubroca and J.L. Feugeas. Moment model hierachy for radiative transferequations. preprint.

[6] B. Dubroca, M. Frank, A. Klar, and G. Thommes. Half space momentapproximation to the radiative heat transfer equations. to appear in ZAMM,2003.

[7] B. Dubroca and A. Klar. Half moment closure for radiative transfer equa-tions. J. Comp. Phys., 180:1–13, 2002.

[8] M. Frank and R. Pinnau. Analysis of a half moment model for radiative heattransfer equations. in preparation.

9

[9] E.M. Gelbard. Simplified spherical harmonics equations and their use inshielding problems. WAPD-T-1182, Bettis Atomic Power Laboratory, 1961.

[10] E. W. Larsen, G. Thommes, and A. Klar. New frequency–averaged approx-imations to the equations of radiative heat transfer. preprint, 2002.

[11] E.W. Larsen, G. Thommes, A. Klar, M. Seaıd, and T. Gotz. Simplified pnapproximations to the equations of radiative heat transfer in glass. J. Comp.Phys., 183:652–675, 2002.

[12] C.D. Levermore. Relating Eddington factors to flux limiters. J. Quant.Spectroscop. Radiat. Transfer, 31:149–160, 1984.

[13] C.D. Levermore and G.C. Pomraning. A flux limited diffusion theory. As-trophysics Journal, 248:321, 1981.

[14] D. Levermore. Moment closure hierarchies for kinetic theories. J. Stat. Phys.,83, 1996.

[15] I. Muller and T. Ruggeri. Rational Extended Thermodynamics. Springer,New York, 1998.

[16] H. Struchtrup. On the number of moments in radiative transfer problems.Annals of Physics, 266:1–26, 1998.

[17] R. Turpault. Construction of a multigroup M1 model for the radiative trans-fer equation. C.R. Acad. Sci. Paris, 334, Serie I:331–336, 2002.

List of Figures

5.1 Incident radiation of the different models for a fixed temperatureprofile in 1D and σ = 1, κ = 1. . . . . . . . . . . . . . . . . . . . . 11

5.2 Incident radiation of the different models for a fixed temperatureprofile in 1D and σ = 0.1,κ = 0.01. . . . . . . . . . . . . . . . . . 12

5.3 Incident radiation of the different models for a fixed temperatureprofile in 2D for σ = 1,κ = 1. Plot along the diagonal x = y. . . . 12

5.4 Incident radiation of the different models for a fixed temperatureprofile in 2D for σ = 0.1, κ = 0.01. Plot along the diagonal x = y. 13

5.5 Incident radiation of the different models for a fixed temperatureprofile in 2D for σ = 0.1, κ = 0.01. Plot along y = 0.5 . . . . . . . 13

5.6 Incident radiation of the different models for discontinuous coeffi-cients κ0 = 3, κ1 = 1. Plot along x = 0.5. . . . . . . . . . . . . . . 14

10

5.7 Incident radiation of the different models for discontinuous coeffi-cients and κ0 = 1, κ1 = 0.1. Plot along y = 5. . . . . . . . . . . . 14

5.8 Incident radiation of the different models for the coupled equationsin 1D with κ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.9 Incident radiation of the different models for the coupled equationsin 1D with κ = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.10 Incident radiation of the different models for the coupled equationsin 2D for κ = 1. Plot along y = 0.5. . . . . . . . . . . . . . . . . . 16

5.11 Incident radiation of the different models for the coupled equationsin 2D for κ = 0.1. Plot along y = 0.5. . . . . . . . . . . . . . . . . 16

5.12 Temperature distribution for a flame–like flow . . . . . . . . . . . 17

5.13 Norm of the relative flux . . . . . . . . . . . . . . . . . . . . . . . 17

Figures

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

5

x

1/4π

∫ I d

Ω

TransportHalf SpaceSP3P1SP2

Figure 5.1: Incident radiation of the different models for a fixed temperatureprofile in 1D and σ = 1, κ = 1.

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9

0.95

1

1.05

1.1

1.15

1.2x 10

5

x

1/4π

∫ I d

Ω

TransportHalf SpaceSP3P1

Figure 5.2: Incident radiation of the different models for a fixed temperatureprofile in 1D and σ = 0.1,κ = 0.01.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

5

x

1/4π

∫ I d

Ω

TransportQuarter SpaceSP3P1

Figure 5.3: Incident radiation of the different models for a fixed temperatureprofile in 2D for σ = 1,κ = 1. Plot along the diagonal x = y.

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

4

6

8

10

12

14

16x 10

4

x

1/4

π ∫ I

dΩ

κ=0.01, σ=0.1

TransportQuarter SpaceSP3P1

Figure 5.4: Incident radiation of the different models for a fixed temperatureprofile in 2D for σ = 0.1, κ = 0.01. Plot along the diagonal x = y.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15.5

6

6.5

7

7.5

8

8.5

9

9.5

10x 10

4

x

1/4

π ∫ I

dΩ

κ=0.01, σ=0.1

TransportQuarter SpaceSP3P1

Figure 5.5: Incident radiation of the different models for a fixed temperatureprofile in 2D for σ = 0.1, κ = 0.01. Plot along y = 0.5

13

0 1 2 3 4 5 6 7 8 9 100.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

5

y

1/4π

∫ I d

Ω

TransportQuarter SpaceSP3P1

Figure 5.6: Incident radiation of the different models for discontinuous coefficientsκ0 = 3, κ1 = 1. Plot along x = 0.5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.55

1.6

1.65

1.7

1.75

1.8

1.85x 10

5

x

1/4π

∫ I d

Ω

TransportQuarter SpaceSP3P1

Figure 5.7: Incident radiation of the different models for discontinuous coefficientsand κ0 = 1, κ1 = 0.1. Plot along y = 5.

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

6

x

∫ I d

Ω

TransportHalf SpaceSP3P1

Figure 5.8: Incident radiation of the different models for the coupled equationsin 1D with κ = 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5x 10

6

x

∫ I d

Ω

Half SpaceTransportSP3P1

Figure 5.9: Incident radiation of the different models for the coupled equationsin 1D with κ = 0.1.

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

6

7

8

9

10

11x 10

4

x

1/4π

∫ I d

Ω

TransportQuarter SpaceSP3P1

Figure 5.10: Incident radiation of the different models for the coupled equationsin 2D for κ = 1. Plot along y = 0.5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15.5

6

6.5

7

7.5

8

8.5

9

9.5

10x 10

4

x

1/4π

∫ I d

Ω

TransportQuarter SpaceSP3P1

Figure 5.11: Incident radiation of the different models for the coupled equationsin 2D for κ = 0.1. Plot along y = 0.5.

16

Figure 5.12: Temperature distribution for a flame–like flow

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

x

|f|

M1 modelP1 model

Figure 5.13: Norm of the relative flux

17