password projet cemracs 2014 financé par chromedespres/chrome/... · e: from the jcp paper n e...

CelineCaldini-

Queiros, B.D. ,

Lise-MarieImbert-Gerard,Maryna

Kachanovska(with OlivierLafitte andRemy Sart)

Passwordprojet Cemracs 2014finance par Chrome

Celine Caldini-Queiros, B. D. , Lise-Marie Imbert-Gerard,Maryna Kachanovska (with Olivier Lafitte and Remy Sart)

Password projet Cemracs 2014 finance par Chrome p. 1 / 24

Introduction

Harmonic 1D

Time domain1D

Section 1

Introduction

Chrome day p. 2 / 24

Introduction

Harmonic 1D

Time domain1D

Context

• General goal : The project Password aimed at studying the numericalsolution of X-Modes equations by means of standard finite elements andfinite difference techniques. X-Mode equations come from the timeharmonic Maxwell’s equations with the cold plasma dielectric tensor. It canbe used to model the heating of a magnetic fusion plasma in Tokamaks(ITER), and is also encountered in reflectometry experiments to probe theplasma.

• This has been performed in 1D.


Introduction

Harmonic 1D

Time domain1D

The cold plasma model

• Linearization of Vlasov-Maxwell for one species=e− (for simplicity)− 1

c2 ∂tE +∇∧ B = µ0J, J = −eNeue ,∂tB +∇∧ E = 0,me∂tue = −e (E + ue ∧ B0)−meνue .

Here ν > 0 is collision frequency : it is a very small quantity in collisionlessplasmas because the plasma has very low collisionality. It corresponds tofriction on a bath of static ions.

• In the frequency domain (∂t = −iω)1c2 iωE +∇∧ B = −µ0eNeue ,−iωB +∇∧ E = 0,−imeωue = −e (E + ue ∧ B0)−meνue .

That isωue + ωc iue ∧ b0 = − e

meiE

where the cyclotron frequency is ωc = e|B0|me

and b0 = B0|B0|

is the normalizedmagnetic field.The complex pulsation is ω = ω + iν.


Introduction

Harmonic 1D

Time domain1D

Dielectric tensor

Therefore ∇∧∇ ∧ E− ω2

c2 εE = 0 with

ε =

1− ωω2

p

ω(ω2−ω2c)

iωcω

2p

ω(ω2−ω2c)

0

−i ωcω2p

ω(ω2−ω2c)

1− ωω2p

ω(ω2−ω2c)

0

0 0 1− ω2p

ωω

, ω2p =

e2Ne

c2ε0c2.

- Assumptions : ω 6= ωc(x) in the domain of study.- The top diagonal coefficient vanishes at the limit ν = 0 for

ω2 = ω2H := ω2

c (x) + ω2p(x).

- PhD thesis Hattori, Post-doc Aurore.


Introduction

Harmonic 1D

Time domain1D

Xmode configuration

curl curlE− ω2

c2(ε0 + iνId) E = 0, ε0(x) =

(α(x) iδ(x)−iδ(x) α(x)

), ν > 0.

Typically

α(x) =

10, x ≤ −10,−x , −10 < x ≤ 5,−5, x > 5.

and

δ(x) =

0, x ≤ −10,

4/30x + 4/3, −10 < x ≤ 5,2, x > 5.

With ω = c = 1 and ∂y = iθ, one gets the strong form−θ2E1 −iθE ′2 − (α(x) + iν)E1 −iδ(x)E2 = 0,iθE ′1 −E ′′2 −iδ(x)E1 − (α(x) + iν)E2 = 0,


Introduction

Harmonic 1D

Time domain1D

Variational formulation

In 1D . Unknowns : u = (E1,E2) = (Ex ,Ey )∫ H

−L

(E ′2 − iθE1)(E ′2 − iθE1)−∫ H

−L

(ε0 + iνId)u · v

−i√α(−L)E2(−L)E2(−L) = −ginc(−L)(E2(−L))

a(u, v) = a1(u, v) + ia2(u, v) and l(v) = −ginc(−L)(v2(−L))a1(u, v) =

∫ H

−L(u′2 − iθu1)(v ′2 − iθv1)−

∫ H

−Lε0u · v,

a2(u, v) = −ν∫ H

−Lu · v −

√α(−L)u2(−L)v2(−L),

where a1 = a∗1 and a2 = a∗2 are hermitian.

u, v ∈ V = (E1,E2)| E1,E2,E′2 ∈ L2

‖u‖2V = ‖E1‖2

L2 + ‖E2‖2L2 + ‖E ′2‖L2


Introduction

Harmonic 1D

Time domain1D

|a(u, u)| ≥ 1

4

√ν

ε+ θ2‖u′2‖2

L2 +ν

4‖u‖2

L2 ,

Coercivity

|a(u, u)| ≥ 1

4

√ν

ε+ θ2‖u′2‖2

L2 +ν

4‖u‖2

L2 .

The problem is well-posed (existence/uniqueness)

Passing to the limit ν → 0+ is the limit absorption principle

- Analysis of the coercivity for the full 3D tensor : PhD thesis Hattori- Passing to the limit in 1D : done in JMPA D.-Imbert-Weder, resonances. . .


Introduction

Harmonic 1D

Time domain1D

Matlab code, P1 FEM (Maryna)

Non singular case : α > 0 everywhere.

(a) Convergence non sigular case


Introduction

Harmonic 1D

Time domain1D

With hybrid resonance : α(0) = 0.

For physically adapted coefficients Ne , B0 hybrid resonance : Eνx ≈ cx+iν

.

E+x = lim

ν→0+Eνx = c

(iπδ + P.V .

1

x

)6∈ L2(Ω).

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−20

−15

−10

−5

0

5

10

15

20

x

ReE

1

ν = 1e− 1, h = 1e− 3ν = 1e− 2, h = 1e− 3ν = 1e− 4, h = 1e− 5

−6 −4 −2 0 2 4 6 8 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

xReE

2

ν = 1e− 1, h = 1e− 3ν = 1e− 2, h = 1e− 3ν = 1e− 4, h = 1e− 5

Convergence Eνx (x) Convergence Eνy (x)

g inc(−L) = −2√

2i exp(−22√

2i), λ =√

10, L = 15, H = 10


Introduction

Harmonic 1D

Time domain1D

Numerical convergence

(b) Ex(x) error (singular case) (c) Ey (x) error (singular case)


Introduction

Harmonic 1D

Time domain1D

(d) hε vs ν (sing.)


Introduction

Harmonic 1D

Time domain1D

Condition number of the block matrix on ν, for several values of h

h-ν 1 0.1 1e-2 1e-3 1e-4 1e-8 0

1 4.987 36.949 42.394 42.757 42.79 42.793 42.793

0.5 48.473 168.41 207.34 209.63 209.83 209.85 209.85

0.1 1146.1 8304.1 19169 20270 20344 20352 20352

0.05 4686.2 38876 1.4e5 1.58e5 1.58e5 1.58e5 1.58e5

0.01 1.2e5 1.1e6 8.3e6 1.8e7 1.93e7 1.93e7 1.93e7

Table : The condition number of the matrix of the system fordifferent values of ν and h.

Remark : the computed matrices are not singular even for ν = 0.


Introduction

Harmonic 1D

Time domain1D

Equations

−ε0∂tEx = −eNe(x)ux ,−ε0∂tEy − ∂xHz = −eNe(x)uy ,∂tHz + ∂xEy = 0,me∂tux = eEx + eB0uy − νux ,me∂tuy = eEy − eB0ux − νuy .

plus boundary conditions on the left hand side.

Scheme coded by Celine (Python, now Fortran).video : ♦


Introduction

Harmonic 1D

Time domain1D

Properties

• The scheme is the 1D version of an extension of the Yee scheme 1.• It preserves the discrete energy equation

Enh = ‖Enh‖2

h + ‖Bn− 12

h ‖2h + ‖Jn

h‖2h −∆t〈En

h, cRBn− 1

2h 〉h.

which is the analogue of the continuous energy

E =

∫Ω

(ε0|E|2

2+|B|2

2µ0+

me |J|2

2|e|Ne

)dx.

1. Stable coupling of the Yee scheme with a linear current model, DaSilva, Campos-Pinto, D, Heuraux, HAL 2014 (to appear in JCP)


Introduction

Harmonic 1D

Time domain1D

”real” Ne : from the JCP paper

Ne Zoom around the foot of the ramp

Cut of the electronic density in the horizontal direction.An additional kink (in red) is sometimes added at x = 500 to evaluate theeffect of an extremely strong gradient.


Introduction

Harmonic 1D

Time domain1D

Another scheme

Full cell-centered : semi lagrangian + splitting

I :

ε0∂tEx = 0,ε0∂tEy + ∂xHz = 0,∂tHz + ∂xEy = 0,me∂tux = 0,me∂tuy = 0.

II :

−ε0∂tEx = −eNe(x)ux ,−ε0∂tEy = −eNe(x)uy ,∂tHz = 0,me∂tux = eEx + eB0uy − νux ,me∂tuy = eEy − eB0ux − νuy .


Introduction

Harmonic 1D

Time domain1D

Semi-lagrangian schemes

• Based on the remark that∂tEy + ∂xHz = 0,∂tHz + ∂xEy = 0,

⇐⇒∂t(Ey + Hz) + ∂x(Ey + Hz) = 0,∂t(Ey − Hz) + ∂x(Ey − Hz) = 0,

and that semi-lagrangian schemes 2 (equivalent to Strang’s stencils)discretize transport with arbitrary order.• The final scheme is co-localised and explicit (n 7→ n + 1)• It satisfies an energy stability inequality under CFL

Enh = ‖Enh‖2

h + ‖Bnh‖2

h + ‖Jnh‖2

h

Enh ≤ En−1h ≤ En−2

h ≤ . . .

2. Widely used in the plasma community


Introduction

Harmonic 1D

Time domain1D

Results : ν = 0.1

T = 2000, order 7 (results very poor for order 1), CFL = 0.5.

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 5 10 15 20 25 30 35 40

’solution’ using 1:3

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35 40


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40


Eνx Ey Hz

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25 30 35 40


-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30 35 40


0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400 1600 1800 2000

’normes.plot’

uνx uy t 7→ E(t)

Limit amplitude OK.


Introduction

Harmonic 1D

Time domain1D

Results : ν = 0.01

-20

-15

-10

-5

0

5

0 5 10 15 20 25 30 35 40


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35 40


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40


Eνx Ey Hz

-4

-3

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40


-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35 40


0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200 1400 1600 1800 2000

’normes.plot’

uνx uy t 7→ E(t)

Limit amplitude OK.


Introduction

Harmonic 1D

Time domain1D

Results : ν = 0.

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

0 5 10 15 20 25 30 35 40


-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35 40


-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35 40


Eνx Ey Hz

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

0 5 10 15 20 25 30 35 40


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35 40


0

50

100

150

200

250

300

350

400

450

500

0 200 400 600 800 1000 1200 1400 1600 1800 2000

’normes.plot’

uνx uy t 7→ E(t)

Ey , Hz and uy have a limit for ν ≈ ν+. Conv. of Ex and ux more delicate toestablish. The stored energy which does not have a limit for t 7→ ∞.


Introduction

Harmonic 1D

Time domain1D

Comparison limit amplitude/limit absorption

Results for ν = 1e − 4 : the theoretical limiting amplitude solution Ey (x) iscompared to the time-dependent solution Ey (x) at t = 79993 andt = 79999 It can be also seen that for ν = 1e − 4 the agreement betweentwo solutions is worse than for ν = 1e − 2.


Introduction

Harmonic 1D

Time domain1D

Comparison limit amplitude/limit absorption

Results for ν = 1e − 2 : we show the L2-error in time and the frequencydomain solution compared to the time domain solution.


Introduction

Harmonic 1D

Time domain1D

General conclusion

Time harmonic : 1D time harmonic solutions converge.

It establishes the numerical limit absorption.

Challenge : coherent 2D and 3D numerical solutions of X-modeequations.

Time domain : The limit amplitude is verified numerically with twodifferent time domain schemes, only for ν > 0.

Challenge : understand the case ν = 0.

Next challenge : moving plasmas.

Numerical analysis of convergence is, of course, difficult for Ex andlow ν.

video Ex video Ey


password projet cemracs 2014 financé par chromedespres/chrome/... · e: from the jcp paper n e...

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