some mathematical aspects of wave propagation in the cold...
TRANSCRIPT
L.-M.Imbert-Gerard
Some mathematical aspects of wave
propagation in the cold plasma model
Lise-Marie Imbert-Gerard,Courant Institute of Mathematical sciences, NYU.
-Collaboartors : B. Despres (UPMC-Paris 6), P. Monk (U.
Delaware) and R. Weder (UNAM, Mexico).
April, 4th 2014.
Some mathematical aspects of wave propagation in the cold plasma model p. 1 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Plan
1 Wave propagation in plasmas
2 Numerical simulation of a cut-offDesign of the basis functionsInterpolation2D simulations
3 About the resonanceA theoretical studyA numerical study
Some mathematical aspects of wave propagation in the cold plasma model p. 2 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Different models
Microscopic approachKinetic approach
1 Kinetic description2 Gyro-kinetic description
Macroscopic approach : Fluid description1 Euler-Maxwell2 Newton-Maxwell
For each species s :
ms (∂tus + us · ∇us) = qs (E + us ∧B) ,
coupled with the Maxwell’s equations
− 1c2 ∂tE + ∇∧ B = µ0J,
∂tB + ∇∧ E = 0, ∇ · B = 0,J =
∑s qsnsus .
Notice that ns has to be provided to close the system.
Some mathematical aspects of wave propagation in the cold plasma model p. 3 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Linearization
Under the hypothesis
E = 0 +E1,B = B0 +B1,us = 0 +us
1,
the system becomes
ms∂tus1 = qs (E1 + us
1 ∧ B0) ,
coupled with
− 1c2 ∂tE1 + ∇∧B1 = µ0J1,
∂tB1 + ∇∧ E1 = 0,J1 =
∑s qsnsu
s1.
The system is closed.
Some mathematical aspects of wave propagation in the cold plasma model p. 4 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The time-harmonic equations
iωc2 E + ∇∧ B = µ0qeneue ,
−iωB + ∇∧ E = 0,−iωmeue = qe (E + ue ∧ B0)
So suppose B0 = |B0|ez , then the Cold Plasma model reads :
∇∧∇ ∧ E =ω2
c2
I −
ω2p
ω2
ω2
ω2−ω2c
i ωωc
ω2−ω2c
0
−i ωωc
ω2−ω2c
ω2
ω2−ω2c
0
0 0 1
E,
where ωc = |qe |B0
meand ω2
p = e2ne (x)ε0me
.
Stix : A general analysis of this model is able to provide a
surprisingly comprehensive view of plasma waves.
Remark In this work ω is such that ω 6= ωc .
Some mathematical aspects of wave propagation in the cold plasma model p. 5 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Classical notation
Define M = I − ω2p
ω2
ω2
ω2−ω2c
i ωωc
ω2−ω2c
0
−i ωωc
ω2−ω2c
ω2
ω2−ω2c
0
0 0 1
= M∗, then
M =
S −iD 0iD S 00 0 P
with S = 12 (R + L), D = 1
2 (R − L) and
R = 1 − ω2p
ω2ω
ω−ωc,
L = 1 − ω2p
ω2ω
ω+ωc,
P = 1 − ω2p
ω2 .
Some mathematical aspects of wave propagation in the cold plasma model p. 6 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Dispersion relation
Wave propagation in the plasmaConsider that the coefficients are locally constant, and look forplane wave solutions
E(x) = e e i(k,x), and e ∈ C3.
Condition for propagation : ω ∈ R for k ∈ R3.
Dispersion relationSuppose k = kn and n = (sin θ, 0, cos θ).Define M(n) = −n ∧ n∧ = I − n ⊗ n, then it reads
det
(M− k2c2
ω2M(n)
)= 0 = A
(k2c2
ω2
)2
− Bk2c2
ω2+ C .
with
A = S sin2 θ + P cos2 θ, B = RL sin2 θ + PS(1 + cos2 θ),
C = detM = PRL.
Some mathematical aspects of wave propagation in the cold plasma model p. 7 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Cut-offs and Resonances
Definition : Cutoff k = 0.Corresponds to PRL = 0.
P = 0 ⇔ ω = ±ωp.
R = 0 ⇔ ω =ωc ±
√ω2
c + 4ω2p
2.
L = 0 ⇔ ω =−ωc ±
√ω2
c + 4ω2p
2.
⇒ ω2p = ω2 + ηωωc , η = −1, 0, 1.
Definition : Resonance k = ∞.Corresponds to A = 0, i.e. tan2 θ = −P
S.
S = 0 for θ =π
2⇔ ω2
p = ω2 − ω2c .
Some mathematical aspects of wave propagation in the cold plasma model p. 8 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Plan
1 Wave propagation in plasmas
2 Numerical simulation of a cut-offDesign of the basis functionsInterpolation2D simulations
3 About the resonanceA theoretical studyA numerical study
Some mathematical aspects of wave propagation in the cold plasma model p. 9 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
O-mode equation and cut-off
Model : wave equation in two dimensions
−∆u + βu = f
smooth varying coefficient
boundary condition : (∂n + iγ) u = g , γ > 0
sign(β) = ±1, Cutoff ⇔ β = 0
β < 0i.e. ne < nc
⇒ Wave propagation
β > 0i.e. ne > nc
⇒ Wave absorption
With nc =c2ε0me
q2e
ω2.
Example : the Airy functions
−u′′ + xu = 0
Some mathematical aspects of wave propagation in the cold plasma model p. 10 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The Ultra Weak Variational formulation,a Plane Wave method
Mesh dependent formulation
Ω = ∪Ωk
Plane wave idea : test functione such that −∆e + βe = 0
∫Ωk
(−∆u + βu)e = 0∫Ωk
(−∆e + βe)u = 0
⇒ ∫
Ωk∇u · ∇e +
∫Ωk
βue −∫∂Ωk
∂νue = 0∫Ωk
∇u · ∇e +∫Ωk
βue −∫∂Ωk
∂νeu = 0
and (∂ν + iγ)u(∂ν + iγ)e − (∂ν − iγ)u(∂ν − iγ)e
= 2iγ(u∂νe + e∂νu)
Some mathematical aspects of wave propagation in the cold plasma model p. 11 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
UWVF Bibliography
B. Despres et al.1994 An ultra-weak-type variational formulation C. R. Acad. Sci.
1998, 2003 with Cessenat on Helmholtz, Maxwell and acoustic waves
2013 with LMIG combining with generalized plane waves, smooth varying coefficients(1)
P. Monk et al.2002, 2004, 2007, 2008, 2012 with Huttunen et al. on Maxwell and elastic waves
2007, 2008, 2012 with Darrigrand combining with FMM
2008 with Buffa on error estimates
R. Hiptmair et al.2009, 2011 with Moiola, Perugia, Gittelson on p and h convergence
2009, 2011 ( Vekua theory )
(1) For some approximation parameter q
−∆ϕ + βqϕ = 0 ⇒ −∆ϕ + βϕ = (β − βq)ϕ= O(hq)
Some mathematical aspects of wave propagation in the cold plasma model p. 12 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DGeneralized plane wave
Case of a constant coefficient
(−∆ − ω2)e iω−→k ·(x ,y) =
(ω2∣∣∣−→k∣∣∣2− ω2
)e iω
−→k ·(x ,y) = 0.
Some mathematical aspects of wave propagation in the cold plasma model p. 13 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DGeneralized plane wave
Case of a constant coefficient
(−∆ − ω2)e iω−→k ·(x ,y) =
(ω2∣∣∣−→k∣∣∣2− ω2
)e iω
−→k ·(x ,y) = 0.
Case of a smooth non constant coefficient
Consider G ∈ R2 s.t. |(x − xG , y − yG )| ≤ h
ϕ(x , y) = eP(x ,y) and P(x , y) =dP∑
i+j=0
λi ,j(x − xG )i (y − yG )j
(−∆ + β)ϕ=((−∂2
xP − ∂2yP − (∂xP)2 − (∂yP)2) + β
)ϕ
⇒ Results in a non linear system⇒ which unknowns are the coefficients λi ,j of P
How to get an explicitly invertible system ?
Some mathematical aspects of wave propagation in the cold plasma model p. 13 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DGeneralized plane wave
Consider a given order q ∈ N∗ such that
β −(∂2
xP + (∂xP)2 + ∂2yP + (∂yP)2
)= O(hq).
Resulting system
λ0,0 = 0 ⇒ ϕ(x , y) bounded
Nun = (dP+1)(dP+2)2 − 1 unknowns
l
k
0
0
d P
d P
Every colored point (k, l) suchthat 0 ≤ k + l ≤ dP
corresponds to the unknownλk,l , coefficient of thepolynomial P .
Some mathematical aspects of wave propagation in the cold plasma model p. 14 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DGeneralized plane wave
Consider a given order q ∈ N∗ such that
β −(∂2
xP + (∂xP)2 + ∂2yP + (∂yP)2
)= O(hq).
Resulting system
λ0,0 = 0 ⇒ ϕ(x , y) bounded
Nun = (dP+1)(dP+2)2 − 1 unknowns
Neq = q(q+1)2 equations
l
k
q-1
q-1
0
0
Every (k, l) belonging to thecolored triangle is such that0 ≤ k + l ≤ q − 1 andcorresponds to the equationthat stems from the coefficient(k, l) of the Taylor expansionup to order q.
Some mathematical aspects of wave propagation in the cold plasma model p. 14 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DGeneralized plane wave
Consider a given order q ∈ N∗ such that
β −(∂2
xP + (∂xP)2 + ∂2yP + (∂yP)2
)= O(hq).
Resulting system
λ0,0 = 0 ⇒ ϕ(x , y) bounded
Nun = (dP+1)(dP+2)2 − 1 unknowns
Neq = q(q+1)2 equations
Choosing the degree of P
dP ≤ q − 1 ⇔ overdetermined system
dP = q ⇒ q equations have no linear term
dP ≥ q + 1 ⇒ underdetermined system
⇒ dP = q + 1 for cheaper computationsand Nun − Neq = 2q + 2
Some mathematical aspects of wave propagation in the cold plasma model p. 14 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DResulting system for the coefficients
∀(i, j) ∈ [[0, q − 1]]2\i + j ≤ q − 1
∂ ix∂j
y β(G)
i !j!= (i + 2)(i + 1)λi+2,j + (j + 2)(j + 1)λi,j+2
+
iX
k=0
jX
l=0
(i − k + 1)(k + 1)λi−k+1,j−lλk+1,l
+
jX
k=0
iX
l=0
(j − k + 1)(k + 1)λi−l,j−k+1λl,k+1.
For a given (i , j) such that0 ≤ i + j ≤ q − 1, the colored(k, l)s represent the λk,l
appearing in the correspondingequation of the system.
Some mathematical aspects of wave propagation in the cold plasma model p. 15 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Basis function design in 2DExplicit solution of the system
∀(i, j) ∈ [[0, q − 1]]2\i + j ≤ q − 1
(i + 2)(i + 1)λi+2,j =∂ i
x∂jy β(G)
i !j!− (j + 2)(j + 1)λi,j+2
−i
X
k=0
jX
l=0
(i − k + 1)(k + 1)λi−k+1,j−lλk+1,l
−
jX
k=0
iX
l=0
(j − k + 1)(k + 1)λi−l,j−k+1λl,k+1.
l
k
q-1
q-1
0
0
q+1
q+1
i
jTO BE COMPUTED
Fixed or
already computed
where x marks represent thecoefficients that are previouslyfixed to compute the rest ofthe coefficients, namely theλk,l such that k = 0, 1.
Some mathematical aspects of wave propagation in the cold plasma model p. 16 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Summary : designing the basis functions
Normalization : Fix λi ,j for i ∈ 0, 1(λ0,1, λ1,0) = N(cos θ, sin θ),N =
√β(G ) or N = i ,
the other coefficients are set to 0 (including λ0,0).
⇒ λi ,j , 0 ≤ i ≤ q + 1, 0 ≤ j ≤ q + 1 − i is explicitly known
The corresponding basis function satisfies(−∆ + βN,θ
q
)ϕ = 0,
whereβN,θ
q = ∂2xP + ∂2
yP + (∂xP)2 + (∂yP)2
Local set of approximated solutions
∀l s.t. 1 ≤ l ≤ p, θl = 2πl/p∀l s.t. 1 ≤ l ≤ p,⇒ E(G , p) = ϕl1≤l≤p
Some mathematical aspects of wave propagation in the cold plasma model p. 17 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Interpolation property of the GPWTheorem, proved thanks to Faa Di Bruno formula
Suppose u is a solution of Helmholtz equation around G ∈ R2.
In addition, suppose that N 6= 0, n ∈ N, q ≥ n + 1, p = 2n + 1and that u satisfies u ∈ Cn+1.Then there are a function ua ∈ SpanE(G , p) depending on αand n and a constant C only depending on α such that ∀X
|u(X ) − ua(X )| ≤ Chn+1 ‖u‖Cn+1 ,
‖∇u(X ) −∇ua(X )‖ ≤ Chn ‖u‖Cn+1 .
Construction ua =
p∑
l=1
xlϕl in O-mode
Since −∆u + βu = 0 and −∆ua + βua = O(hq),
consider the Taylor expansion of u − ua,
⇒ xl , 1 ≤ l ≤ p is the solution of an explicit linear system.
Some mathematical aspects of wave propagation in the cold plasma model p. 18 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Exact solution and its interpolationβ(x , y) = x − 1 and u = Airy(x)e iy
Around G = (−3, 1), approximation computed with p = 7 basisfunctions, at the order q = 4.
Some mathematical aspects of wave propagation in the cold plasma model p. 19 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
h convergence at G = (−3, 1), p = 7 and q = 4
|u(X ) − ua(X )| ≤ Chn+1 ‖u‖Cn+1 ,n = 3, p = 2n + 1 and q = n + 1.
Consider the error as maxℜ(u − ua) and n = 3 :⇒ expected convergence rate n + 1 = 4.
h 1/2 1/22 1/23 1/24 1/25 1/26
error 7.5e-4 4.6e-5 2.7e-6 1.6e-7 1.0e-8 6.2e-10rate - 4.04 4.08 4.05 4.02 4.01
error PW 1.1e-2 4.1e-3 1.9e-3 9.2e-4 4.6e-4 2.3e-4rate PW - 1.41 1.12 1.03 1.01 1.00
Some mathematical aspects of wave propagation in the cold plasma model p. 20 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Behaviour of the basis functions around the cutoff,p = 7 and q = 4.
Getting closer to the cutoff
From the propagative zone G = (1 − h, 1)
h 1/2 1/22 1/23 1/24 1/25 1/26
error 3.2e-4 2.1e-5 1.4e-6 8.9e-8 5.7e-9 3.6e-10rate - 3.91 3.94 3.97 3.97 3.98
error PW 3.8e-3 4.1e-4 5.9e-5 1.1e-5 2.2e-6 5.0e-7rate PW - 3.23 2.78 2.47 2.27 2.15
From the non propagative zone G = (1 + h, 1)
h 1/2 1/22 1/23 1/24 1/25 1/26
error 3.2e-4 2.2e-5 1.4e-6 9.1e-8 5.8e-9 3.7e-10rate - 3.88 3.93 3.96 3.98 3.98
Some mathematical aspects of wave propagation in the cold plasma model p. 21 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
2D Code
Based on the Ultra Weak Variational Formulation
A Method based on the skeleton of the meshCoupling at the interface between cellsNo exact integral formulaApproximated solutions + quadrature formulasimplemented for smooth non constant coefficients
Collaboration with Peter Monk, University of Delaware
Coded in Matlab
Linear system solved with Matlab \ command
Toy code, with good performances nevertheless
Typically :
the mesh has up to 30000 elements, the size of the matrices is 250000× 250000, the computing time is 3 hours.
Some mathematical aspects of wave propagation in the cold plasma model p. 22 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
2D Code
First numerical result,proposed by O. Maj, Max-Planck-Institut fur Plasmaphysik, Garching, Germany.
Exact solution ue(x , y) = Airy(x)e iy ,
solution of Helmholtz equation −∆u + (x − 1)u = 0.
Some mathematical aspects of wave propagation in the cold plasma model p. 23 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Code 2D : convergence results
Approximation of ue(x , y) = Airy(x)e iy ,using Weddle quadrature formula on [−6, 3] × [−1, 1].
100
101
102
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
L2 error at the center of the mesh cells
p=3, q=2p=3, q=3p=3, q=4p=3, q=5p=3, q=6p=5, q=2p=5, q=3p=5, q=4p=5, q=5p=5, q=6p=7, q=2p=7, q=3p=7, q=4p=7, q=5p=7, q=6p=9, q=2p=9, q=3p=9, q=4p=9, q=5p=9, q=6
Some mathematical aspects of wave propagation in the cold plasma model p. 24 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Examples of numerical results obtained withgeneralized PW
A reflectometry test caseApproximation of Helmholtz equationwith a smooth coefficient vanishing along the line x = 4
Some mathematical aspects of wave propagation in the cold plasma model p. 25 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Plan
1 Wave propagation in plasmas
2 Numerical simulation of a cut-offDesign of the basis functionsInterpolation2D simulations
3 About the resonanceA theoretical studyA numerical study
Some mathematical aspects of wave propagation in the cold plasma model p. 26 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Focus on a typical X mode equation
Model equation in two dimensions
Introducing the vorticity W the system reads
W +∂yEx −∂xEy = 0,∂yW −αEx −iγEy = 0,
−∂xW +iγEx −αEy = 0.Resonance at α = 0
The Budden problem example : a singular solution
For α(x , y) = −x
For γ(x , y) =√
x2 − x/4 + 1,Ex is not integrable at the resonance
Ey (x , y) = −ex/2 + xe−x/2Ei(x),
Ex(x , y) = i
√x2− x
4+1
xEy .
Some mathematical aspects of wave propagation in the cold plasma model p. 27 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
A simpler problem
x0 H−L
y RESONANCE
COERCIVE ZONE
WALL
Real geometry Simplified geometrycourtesy of S. Heuraux
W +iθU −(V )′ = 0iθW −(α(x) + iν)U −iγ(x)V = 0−(W )′ +iγ(x)U −(α(x) + iν)V = 0
Some mathematical aspects of wave propagation in the cold plasma model p. 28 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Different points of view
W +iθU −(V )′ = 0iθW −(α(x) + iν)U −iγ(x)V = 0−(W )′ +iγ(x)U −(α(x) + iν)V = 0
A Cauchy problem
d
dx
(V
W
)= Aν(x)
(V
W
)
where Aν(x) =
(θγ(x)
α(x)+iν 1 − θ2
α(x)+iνγ(x)2
α(x)+iν − α(x) − iν − θγ(x)α(x)+iν
)
An integral equation
(α(x) + iν)U(x) −∫ x
G
K ν(x , z)U(z)dz = F νG (x),
Some mathematical aspects of wave propagation in the cold plasma model p. 29 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The integral point of view
Integral equations in 1D
First kind ind
∫K (x , y)u(y)dy = f (x)
Second kind u(x) −∫
K (x , y)u(y)dy = f (x)
Third kind id g(x)u(x) −∫
K (x , y)u(y)dy = f (x).
See Picard (1911) and Hilbert (1956) for g(x) = x .
⇒ See Bart-Warnock (1973) for K (0, 0) = 0 and g(0) = 0.
u(x) = w0δ(x) + V .P .
[ϕ(x)
g(x)
].
Non uniqueness since the Dirac function δ(t) is in thekernel of the homogeneous operator.
Some mathematical aspects of wave propagation in the cold plasma model p. 30 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
A first estimate. . . satisfyingwhich is not satisfactory
Consider G = H :
(α(x) + iν)U(x)−∫ x
H
K (x , z)U(z) = FH(x), ‖FH‖∞ ≤ C .
Then : (r2x2 + ν2) |U(x)| ≤ ‖FH‖∞ +
∫ H
x
|K (x , z)||U(z)|dz
Since for 0 ≤ x ≤ z ‖K (x , z)‖L∞ ≤ β|x − z |Since for 0 ≤ x ≤ z ‖K (x , z)‖L∞ ≤ βz
Since for 0 ≤ x ≤ z : ‖K (x , z)‖L∞ ≤ β√
r2z2 + ν2 , then
√r2x2 + ν2|U(x)| ≤ c‖FH‖∞ + β
∫ H
x
√r2z2 + ν2|U(z)|dz .
Gronwall ⇒ |U(x)| ≤ C
r2x2 + ν2, UNBOUNDED as ν → 0.
Some mathematical aspects of wave propagation in the cold plasma model p. 31 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Define two convenient basis solutions
Idea : define two basis functions focusing on
their behavior at x = 0⇒ Linear independence
their behavior at x = ∞⇒ Distinguish a physical solution
The first basis function
U1(0) = 0+ a normalization for the two other components
exponentially growing at large scale
The second basis function : the physical one
exponentially decreasing at large scale
U2(0) = 1iν
Some mathematical aspects of wave propagation in the cold plasma model p. 32 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
A second estimate : a heating estimateA convenient tool
Q(U) = V1(H)W (H) − W1(H)V (H).
This quantity is the Wronskian of the current solution U andthe first basis function.It is independent of the position H.
Proposition : There existsa constant Cθ (with continuous dependence with respect to θ)and a continuous function ν 7→ ǫ(ν) with ǫ(0) = 0 such that
∣∣∣∣ |ν| ‖U‖2L2(−L,H) −
∣∣∣∣πQ(U)2
α′(0)
∣∣∣∣∣∣∣∣ ≤ Cθǫ(ν)‖H‖2.
U2 is the physical solution and Q(U2) = 1
⇒ The heating of the ions isπ
|α′(0)| .
Some mathematical aspects of wave propagation in the cold plasma model p. 33 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Explicit singular solutions
Theorem
U2 → U,±2 =
(P .V .
1
α(x)± iπ
α′(0)δD + u±
2 , v±2 , w±
2
)
where u±2 , v±
2 ,w±2 ∈ L2(−L,∞)
and δD is the Dirac mass at the origin.
The limits U±2 are solutions in the sense of distributions. They
will be called the singular solutions.
Back to Bart-Warnock
Non uniqueness of the solution of the initial equation
α(x)Uν=0(x) −∫ x
G
K ν=0(x , z)Uν=0(z)dz = F ν=0G (x),
Some mathematical aspects of wave propagation in the cold plasma model p. 34 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Numerically approximated system : ν 6= 0
d
dx
(V
W
)= Aν(x)
(V
W
)
where Aν(x) =
(θγ(x)
α(x)+iν 1 − θ2
α(x)+iνγ(x)2
α(x)+iν − α(x) − iν − θγ(x)α(x)+iν
)
together with U = 1α(x)+iν (iθW − iγ(x)V )
x = −L x = H
x
α(x)
slope −r
γ(x)
Some mathematical aspects of wave propagation in the cold plasma model p. 35 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The first basis functionθ = 0, ν = 10−2
For any bounded θ : θ ∈ [θ−, θ+],
∥∥∥∥(
U1(x) − F ν(x)
α(x) + iν
)−(Uν=0
1 − F ν=0)(x)
∥∥∥∥L∞(]−L,H[)
→ 0
U1 converges in L∞loc(] − L, 0[∪]0,H[)
V1 and W1 converge in L∞(] − L,H[)
Some mathematical aspects of wave propagation in the cold plasma model p. 36 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The first basis functionθ = 0, ν = 10−2
For any interval bounded θ,∥∥∥∥(
U1(x) − F ν(x)
α(x) + iν
)−(Uν=0
1 − F ν=0)(x)
∥∥∥∥L∞(]−L,H[)
→ 0
U1 converges in L∞loc(] − L, 0[∪]0,H[)V1 and W1 converge in L∞(] − L,H[)
−10 −5 0 5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1 µ=10−2
Re v1Re w1Re u1
−10 −5 0 5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
µ=10−2
Im v1Im w1Im u1
Some mathematical aspects of wave propagation in the cold plasma model p. 37 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The first basis functionθ = 0, ν = 10−3
For any interval bounded θ,∥∥∥∥(
U1(x) − F ν(x)
α(x) + iν
)−(Uν=0
1 − F ν=0)(x)
∥∥∥∥L∞(]−L,H[)
→ 0
U1 converges in L∞loc(] − L, 0[∪]0,H[)V1 and W1 converge in L∞(] − L,H[)
−10 −5 0 5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1 µ=10−3
Re v1Re w1Re u1
−10 −5 0 5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
µ=10−3
Im v1Im w1Im u1
Some mathematical aspects of wave propagation in the cold plasma model p. 37 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The first basis functionθ = 0, ν = 10−4
For any interval bounded θ,∥∥∥∥(
U1(x) − F ν(x)
α(x) + iν
)−(Uν=0
1 − F ν=0)(x)
∥∥∥∥L∞(]−L,H[)
→ 0
U1 converges in L∞loc(] − L, 0[∪]0,H[)V1 and W1 converge in L∞(] − L,H[)
−10 −5 0 5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1 µ=10−4
Re v1Re w1Re u1
−10 −5 0 5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
µ=10−4
Im v1Im w1Im u1
Some mathematical aspects of wave propagation in the cold plasma model p. 37 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 1.5, ν = 10−2
For any bounded θ,∃C , Cp independent of ν s. t.
|V2(H)| + |W2(H)| ≤ C .
∥∥∥∥U2 −1
α(·) + iν
∥∥∥∥Lp(−L,H)
≤ Cp, 1 ≤ p < ∞.
Some mathematical aspects of wave propagation in the cold plasma model p. 38 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 1.5, ν = 10−2
For any bounded θ,∃C , Cp independent of ν s. t. |V2(H)| + |W2(H)| ≤ C ,
∥∥∥∥U2 −1
α(·) + iν
∥∥∥∥Lp(−L,H)
≤ Cp, 1 ≤ p < ∞.
−10 −5 0 5−60
−40
−20
0
20
40
60µ=10−2
Re v2Re w2Re u2Re 1/(−x+imu)
−10 −5 0 5−60
−40
−20
0
20
40
60µ=10−2
Re v2Re w2Re u2
Some mathematical aspects of wave propagation in the cold plasma model p. 39 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 1.5, ν = 10−3
For any bounded θ,∃C , Cp independent of ν s. t. |V2(H)| + |W2(H)| ≤ C ,
∥∥∥∥U2 −1
α(·) + iν
∥∥∥∥Lp(−L,H)
≤ Cp, 1 ≤ p < ∞.
−10 −5 0 5−600
−400
−200
0
200
400
600µ=10−3
Re v2Re w2Re u2Re 1/(−x+imu)
−10 −5 0 5−60
−40
−20
0
20
40
60µ=10−3
Re v2Re w2Re u2
Some mathematical aspects of wave propagation in the cold plasma model p. 39 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 1.5, ν = 10−4
For any bounded θ,∃C , Cp independent of ν s. t. |V2(H)| + |W2(H)| ≤ C ,
∥∥∥∥U2 −1
α(·) + iν
∥∥∥∥Lp(−L,H)
≤ Cp, 1 ≤ p < ∞.
−10 −5 0 5−6000
−4000
−2000
0
2000
4000
6000µ=10−4
Re v2Re w2Re u2Re 1/(−x+imu)
−10 −5 0 5−60
−40
−20
0
20
40
60µ=10−4
Re v2Re w2Re u2
Some mathematical aspects of wave propagation in the cold plasma model p. 39 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionpositive and negative regularization
U+2 (x) = U−
2 (x) 0 < x .
U+2 (x) − U−
2 (x) =−2iπ
α′(0)Uν=0
1 (x) x < 0.
Some mathematical aspects of wave propagation in the cold plasma model p. 40 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 0, ν = ±10−2
U+2 (x) = U−
2 (x) 0 < x .
U+2 (x) − U−
2 (x) =−2iπ
α′(0)Uν=0
1 (x) x < 0.
−10 −5 0 5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1µ= ± 10−2
Re u1Im (u2+ − u2−)/2pi
Some mathematical aspects of wave propagation in the cold plasma model p. 40 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 0, ν = ±10−3
U+2 (x) = U−
2 (x) 0 < x .
U+2 (x) − U−
2 (x) =−2iπ
α′(0)Uν=0
1 (x) x < 0.
−10 −5 0 5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1µ= ± 10−3
Re u1Im (u2+ − u2−)/2pi
Some mathematical aspects of wave propagation in the cold plasma model p. 41 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
The second basis functionθ = 0, ν = ±10−4
U+2 (x) = U−
2 (x) 0 < x .
U+2 (x) − U−
2 (x) =−2iπ
α′(0)Uν=0
1 (x) x < 0.
−10 −5 0 5−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1µ= ± 10−4
Re u1Im (u2+ − u2−)/2pi
Some mathematical aspects of wave propagation in the cold plasma model p. 41 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
On going work
O-X mode conversionExpansion in the mode conversionregion
(∂x + d∂y )E = xE + yB
(∂x + d∂y )B = xE − xB
Weak FormulationZ
Ω
„
1 dr
dr |d|2
« „
∂xu
∂y u
«
·
„
∂x v
∂y v
«
− 2idi
Z
Ωx∂y uv
+
Z
Ω
„
1 +1
µ+ x(x + y)
«
uv + iσ
Z
∂Ωuv =
Z
∂Ωgv.
Some mathematical aspects of wave propagation in the cold plasma model p. 42 / 43
L.-M.Imbert-Gerard
Wavepropagationin plasmas
Numericalsimulation ofa cut-off
Design of thebasis functions
Interpolation
2D simulations
About theresonance
A theoreticalstudy
A numericalstudy
Thanks for your attention.
Some mathematical aspects of wave propagation in the cold plasma model p. 43 / 43