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Perfectly Matched Layers for Maxwell Equations in Plasmas

• E. B´

ecache, P. Joly, M. Kachanovska

POEMS, INRIA, ENSTA ParisTech

December 3, 2014

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 1 / 32

Outline

Outline

1 Introduction

Maxwell Equations in Cold Plasma Model Problems: Isotropic Dispersive Model and Uniaxial Plasma Model (L.Colas) Introduction into the PML

2 PMLs for Plasmas

Drude Model for Metamaterials: General Dispersive Isotropic Model Uniaxial Cold Plasma Model in 2D Uniaxial Cold Plasma Model in 3D

3 Conclusions

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 2 / 32

Introduction Maxwell Equations in Cold Plasma

Model Problem

B0 = (0, 0, B0) x y z Maxwell Equations in the Frequency Domain in R3 In the frequency domain in R3 − curl curl ˆ E + ω2 c2 ǫ(ω)ˆ E = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 3 / 32

Introduction Maxwell Equations in Cold Plasma

Model Problem

B0 = (0, 0, B0) x y z Maxwell Equations in the Frequency Domain in R3 In the frequency domain in R3 − curl curl ˆ E + ω2 c2 ǫ(ω)ˆ E = 0. Cold plasma dielectric tensor: ǫ(ω) =       1 −

ω2

p

ω2−ω2

c

−i

ω2

pωc

ω(ω2−ω2

c )

i

ω2

pωc

ω(ω2−ω2

c )

1 −

ω2

p

ω2−ω2

c

1 −

ω2

p

ω2

      Single-species cold plasma Ne(x, y, z) concentration of particles e particle charge me particle mass ωp =

• Nee2

mǫ0

plasma frequency ωc = eB0

mec

algebraic cyclotron frequency

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 3 / 32

Introduction Maxwell Equations in Cold Plasma

Two limit cases

ǫ(ω) =       1 −

ω2

p

ω2−ω2

c

−i

ω2

pωc

ω(ω2−ω2

c )

i

ω2

pωc

ω(ω2−ω2

c )

1 −

ω2

p

ω2−ω2

c

1 −

ω2

p

ω2

     

1

ωc = 0 (i.e. B0 = 0) ǫ(ω) =      1 −

ω2

p

ω2

1 −

ω2

p

ω2

1 −

ω2

p

ω2

     =

• 1 −

ω2

p

ω2

• I3

This is an isotropic dispersive case.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 4 / 32

Introduction Maxwell Equations in Cold Plasma

Two limit cases

ǫ(ω) =       1 −

ω2

p

ω2−ω2

c

−i

ω2

pωc

ω(ω2−ω2

c )

i

ω2

pωc

ω(ω2−ω2

c )

1 −

ω2

p

ω2−ω2

c

1 −

ω2

p

ω2

     

1

ωc = 0 (i.e. B0 = 0) ǫ(ω) =      1 −

ω2

p

ω2

1 −

ω2

p

ω2

1 −

ω2

p

ω2

     =

• 1 −

ω2

p

ω2

• I3

This is an isotropic dispersive case.

2

ωc → ∞ ǫ(ω) =    1 1 1 −

ω2

p

ω2

   This is an anisotropic dispersive case (uniaxial plasma (L. Colas))

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 4 / 32

Introduction Model Problems

Two Problems Recast in the Time Domain

Generalized hyperbolic system: ∂tU +

Anisotropy

• d
• i=1

Ai∂xi U +

Anisotropy+Dispersion

• BU

= 0, U(x, t) ∈ Rm, Ai, B ∈ Rm×m, i = 1, . . . , d.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 5 / 32

Introduction Model Problems

Two Problems Recast in the Time Domain

Generalized hyperbolic system: ∂tU +

Anisotropy

• d
• i=1

Ai∂xi U +

Anisotropy+Dispersion

• BU

= 0, U(x, t) ∈ Rm, Ai, B ∈ Rm×m, i = 1, . . . , d. U → U0ei(ωt−k·x) = ⇒ FA,B (ω, k) = 0 = ⇒ ω : ℑω = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 5 / 32

Introduction Model Problems

Two Problems Recast in the Time Domain

Generalized hyperbolic system: ∂tU +

Anisotropy

• d
• i=1

Ai∂xi U +

Anisotropy+Dispersion

• BU

= 0, U(x, t) ∈ Rm, Ai, B ∈ Rm×m, i = 1, . . . , d. U → U0ei(ωt−k·x) = ⇒ FA,B (ω, k) = 0 = ⇒ ω : ℑω = 0. We rescale the equations so that c = 1 = ǫ0 = µ0. ωc = 0 (isotropic dispersive) ∂tE − curl B = −J, ∂tB + curl E = 0, ∂tJ = ω2

pE.

ωc → ∞ (anisotropic dispersive) ∂tE − curl B = −jpe3, ∂tB + curl E = 0, ∂tjp = ω2

pEz.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 5 / 32

Introduction Introduction into the PML

Maxwell Equations in Plasma: Dealing with Unbounded Domains

The PML approach deals with the unboundedness of the domain Physical Domain

PML (non-physical medium)

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32

Introduction Introduction into the PML

Maxwell Equations in Plasma: Dealing with Unbounded Domains

The PML approach deals with the unboundedness of the domain Physical Domain

PML (non-physical medium)

Properties of the PML

1

’Perfect matching’, i.e. zero reflections at the interface between the physical domain and the PML

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32

Introduction Introduction into the PML

Maxwell Equations in Plasma: Dealing with Unbounded Domains

The PML approach deals with the unboundedness of the domain Physical Domain

PML (non-physical medium)

Properties of the PML

1

’Perfect matching’, i.e. zero reflections at the interface between the physical domain and the PML

2

Inside the PML the solution decays exponentially fast, so that on the PML boundary zero Dirichlet or Neumann BCs can be posed

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32

Introduction Introduction into the PML

Maxwell Equations in Plasma: Dealing with Unbounded Domains

The PML approach deals with the unboundedness of the domain Physical Domain

PML (non-physical medium)

Properties of the PML

1

’Perfect matching’, i.e. zero reflections at the interface between the physical domain and the PML

2

Inside the PML the solution decays exponentially fast, so that on the PML boundary zero Dirichlet or Neumann BCs can be posed

3

Non-physical/anisotropic phenomenon

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32

Introduction Introduction into the PML

Standard PML for Maxwell Equations

B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables (non-split formulation).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32

Introduction Introduction into the PML

Standard PML for Maxwell Equations

B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables (non-split formulation). PML in the direction n n

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32

Introduction Introduction into the PML

Standard PML for Maxwell Equations

B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables (non-split formulation). PML in the direction n n One way to write a PML system (PML in ex-direction): ex

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32

Introduction Introduction into the PML

Standard PML for Maxwell Equations

B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables (non-split formulation). PML in the direction n n One way to write a PML system (PML in ex-direction): ex

1

rewrite the equations in the frequency domain (∂t → iω)

2

perform a change of variables ˜ x := x +

1 iω x

• σ(x′)dx′, with

σ(x′) > 0 for x > 0 (analytic continuation)

3

come back to the time-domain Re x Im x

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32

Introduction Introduction into the PML

Standard PML for Maxwell Equations

B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables (non-split formulation). PML in the direction n n One way to write a PML system (PML in ex-direction): ex

1

rewrite the equations in the frequency domain (∂t → iω)

2

perform a change of variables ˜ x := x +

1 iω x

• σ(x′)dx′, with

σ(x′) > 0 for x > 0 (analytic continuation)

3

come back to the time-domain Re x Im x Attenuates the amplitude of modes propagating in the direction ex. Other directions: similarly, in corners: multiple changes of variables.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32

Introduction Introduction into the PML

Stability of the B´ erenger’s Perfectly Matched Layers

Anisotropic system: ∂tU +

d

• i=1

Ai∂xi U = 0. U = U0ei(ωt−k·x), fix k, look at ω(k). F(ω, k) = 0 the dispersion relation of a generalized hyperbolic system Vph = ω

k k k

phase velocity (ω = ω(k) solves the dispersion relation) Vg = ∂ω

∂k

group velocity

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 8 / 32

Introduction Introduction into the PML

Stability of the B´ erenger’s Perfectly Matched Layers

Anisotropic system: ∂tU +

d

• i=1

Ai∂xi U = 0. U = U0ei(ωt−k·x), fix k, look at ω(k). F(ω, k) = 0 the dispersion relation of a generalized hyperbolic system Vph = ω

k k k

phase velocity (ω = ω(k) solves the dispersion relation) Vg = ∂ω

∂k

group velocity High-Frequency Stability Criterion for Anisotropic Non-Dispersive Medium (E. B´ ecache et al. 2003) A necessary condition of the stability of the PML in the direction n is (Vf · n) (Vg · n) ≥ 0 (i.e. there are no backward propagating waves).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 8 / 32

Introduction Introduction into the PML

Stability of the B´ erenger’s Perfectly Matched Layers

Anisotropic system: ∂tU +

d

• i=1

Ai∂xi U = 0. U = U0ei(ωt−k·x), fix k, look at ω(k). F(ω, k) = 0 the dispersion relation of a generalized hyperbolic system Vph = ω

k k k

phase velocity (ω = ω(k) solves the dispersion relation) Vg = ∂ω

∂k

group velocity High-Frequency Stability Criterion for Anisotropic Non-Dispersive Medium (E. B´ ecache et al. 2003) A necessary condition of the stability of the PML in the direction n is (Vf · n) (Vg · n) ≥ 0 (i.e. there are no backward propagating waves). A potential source of instability of the PMLs is pure anisotropy, i.e. |Vf | = f

• k

k

• .
• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 8 / 32

Introduction Introduction into the PML

Stability of the B´ erenger’s Perfectly Matched Layers

Anisotropic system: ∂tU +

d

• i=1

Ai∂xi U+BU = 0. U = U0ei(ωt−k·x), fix k, look at ω(k). F(ω, k) = 0 the dispersion relation of a generalized hyperbolic system Vph = ω

k k k

phase velocity (ω = ω(k) solves the dispersion relation) Vg = ∂ω

∂k

group velocity High-Frequency Stability Criterion for Anisotropic Non-Dispersive Medium (E. B´ ecache et al. 2003) A necessary condition of the stability of the PML in the direction n is (Vf · n) (Vg · n) ≥ 0 (i.e. there are no backward propagating waves). A potential source of instability of the PMLs is pure anisotropy, i.e. |Vf | = f

• k

k

• .

Backward propagating waves: dispersive models (|Vf | = f

• k, k

k

• (E. B´

ecache et al., work in progress)).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 8 / 32

PMLs for Plasmas Drude Model

Drude Model for Metamaterials (B0 = 0) (E. B´ ecache, P. Joly, V. Vinoles)

Maxwell equations in the frequency domain with ǫ(ω) =

• 1 − ω2

e

ω2

• ,

µ(ω) =

• 1 − ω2

m

ω2

• .

E = (Ey, Ez)T .

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 9 / 32

PMLs for Plasmas Drude Model

Drude Model for Metamaterials (B0 = 0) (E. B´ ecache, P. Joly, V. Vinoles)

Maxwell equations in the frequency domain with ǫ(ω) =

• 1 − ω2

e

ω2

• ,

µ(ω) =

• 1 − ω2

m

ω2

• .

E = (Ey, Ez)T . Drude Model ∂tE − curl Bx + ω2

eJ = 0,

∂tBx + curl E + ω2

mKx = 0,

∂tJ = E, ∂tKx = Bx. 2D Isotropic Plasma (B0 = 0) ∂tE − curl Bx + ω2

pJ = 0,

∂tBx + curl E = 0, ∂tJ = E. Drude model with ωm = 0, ωe = ωp.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 9 / 32

PMLs for Plasmas Drude Model

Drude Model for Metamaterials (B0 = 0) (E. B´ ecache, P. Joly, V. Vinoles)

Maxwell equations in the frequency domain with ǫ(ω) =

• 1 − ω2

e

ω2

• ,

µ(ω) =

• 1 − ω2

m

ω2

• .

E = (Ey, Ez)T . Drude Model ∂tE − curl Bx + ω2

eJ = 0,

∂tBx + curl E + ω2

mKx = 0,

∂tJ = E, ∂tKx = Bx. 2D Isotropic Plasma (B0 = 0) ∂tE − curl Bx + ω2

pJ = 0,

∂tBx + curl E = 0, ∂tJ = E. Drude model with ωm = 0, ωe = ωp. Dispersion Relation Fk(ω) = ω4 − (ω2

e + ω2 m + k2)ω2 + ω2 eω2 m.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 9 / 32

PMLs for Plasmas Drude Model

Solutions of the Dispersion Relation and Backward Waves

Assume ωe > ωm, let Fk(±ω1,2) = 0. ωm ωe ω2 ω1 k

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 10 / 32

PMLs for Plasmas Drude Model

Solutions of the Dispersion Relation and Backward Waves

Assume ωe > ωm, let Fk(±ω1,2) = 0. ωm ωe ω2 ω1 k Case ω < ωm: all waves propagate only backwards ∂ω1 ∂kj ω1 k kj k ≤ 0, j = 1, 2.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 10 / 32

PMLs for Plasmas Drude Model

Solutions of the Dispersion Relation and Backward Waves

Assume ωe > ωm, let Fk(±ω1,2) = 0. ωm ωe ω2 ω1 k Case ω < ωm: all waves propagate only backwards ∂ω1 ∂kj ω1 k kj k ≤ 0, j = 1, 2. Case ω > ωe: all waves propagate only forward ∂ω2 ∂kj ω2 k kj k ≥ 0, j = 1, 2.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 10 / 32

PMLs for Plasmas Drude Model

Solutions of the Dispersion Relation and Backward Waves

Assume ωe > ωm, let Fk(±ω1,2) = 0. ωm ωe ω2 ω1 k Case ω < ωm: all waves propagate only backwards ∂ω1 ∂kj ω1 k kj k ≤ 0, j = 1, 2. Case ω > ωe: all waves propagate only forward ∂ω2 ∂kj ω2 k kj k ≥ 0, j = 1, 2. B´ erenger PML: unstable for the Drude model if ωm > 0 stable for the isotropic plasma model (ωm = 0).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 10 / 32

PMLs for Plasmas Drude Model

Stability of the B´ erenger’s PML for the isotropic dispersive plasma model

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 11 / 32

PMLs for Plasmas Drude Model

Instability of the B´ erenger’s PMLs for the Drude model

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 12 / 32

PMLs for Plasmas Drude Model

Dealing with the instability of the PML

Idea (S. Cummer 2004, generalization and analysis by E. B´ ecache, P. Joly and V. Vinoles (in preparation)) Instead of x → x +

1 iω x

• σ(x′)dx′ use x → x +

1 iωψ(ω) x

• σ(x′)dx′.
• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 13 / 32

PMLs for Plasmas Drude Model

Dealing with the instability of the PML

Idea (S. Cummer 2004, generalization and analysis by E. B´ ecache, P. Joly and V. Vinoles (in preparation)) Instead of x → x +

1 iω x

• σ(x′)dx′ use x → x +

1 iωψ(ω) x

• σ(x′)dx′.

Uniform Stability Criterion (E. B´ ecache, P. Joly and V. Vinoles) If the PML system is uniformly stable (i.e. stable for all k ∈ R2 and σ > 0), and ψ(ω) : R → R, then ψ(ω) (Vf · n) (Vg · n) ≥ 0. ωm ω∗ ωe ψ(ω) > 0 ψ(ω) < 0 k

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 13 / 32

PMLs for Plasmas Drude Model

Dealing with the instability of the PML

Idea (S. Cummer 2004, generalization and analysis by E. B´ ecache, P. Joly and V. Vinoles (in preparation)) Instead of x → x +

1 iω x

• σ(x′)dx′ use x → x +

1 iωψ(ω) x

• σ(x′)dx′.

Uniform Stability Criterion (E. B´ ecache, P. Joly and V. Vinoles) If the PML system is uniformly stable (i.e. stable for all k ∈ R2 and σ > 0), and ψ(ω) : R → R, then ψ(ω) (Vf · n) (Vg · n) ≥ 0. ωm ω∗ ωe ψ(ω) > 0 ψ(ω) < 0 k One of the choices: ψ(ω) = 1 − ω2

∗

ω2 , ω∗ ∈ [ωm, ωe] (rational in ω; as ω → +∞, ψ(ω) → 1).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 13 / 32

PMLs for Plasmas Drude Model

Stable PML for the Drude Model

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 14 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Two-dimensional uniaxial cold plasma model

Maxwell Equations in Plasma ∂tBx + ∂yEz − ∂zEy = 0, ∂tEy − ∂zBx = 0, ∂tEz + ∂yBx + jp = 0, ∂tjp = ω2

pEz.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 15 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Two-dimensional uniaxial cold plasma model

Maxwell Equations in Plasma ∂tBx + ∂yEz − ∂zEy = 0, ∂tEy − ∂zBx = 0, ∂tEz + ∂yBx + jp = 0, ∂tjp = ω2

pEz.

The dispersion relation: k2

z

ω2 + k2

y

ω2

• 1 −

ω2

p

ω2

= 1.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 15 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Two-dimensional uniaxial cold plasma model

Maxwell Equations in Plasma ∂tBx + ∂yEz − ∂zEy = 0, ∂tEy − ∂zBx = 0, ∂tEz + ∂yBx + jp = 0, ∂tjp = ω2

pEz.

The dispersion relation: k2

z

ω2 + k2

y

ω2

• 1 −

ω2

p

ω2

= 1. Slowness curves (sy = ky

ω , sz = kz ω ):

s2

z +

s2

y

• 1 −

ω2

p

ω2

= 1.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 15 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Two-dimensional uniaxial cold plasma model

Maxwell Equations in Plasma ∂tBx + ∂yEz − ∂zEy = 0, ∂tEy − ∂zBx = 0, ∂tEz + ∂yBx + jp = 0, ∂tjp = ω2

pEz.

The dispersion relation: k2

z

ω2 + k2

y

ω2

• 1 −

ω2

p

ω2

= 1. Slowness curves (sy = ky

ω , sz = kz ω ):

s2

z +

s2

y

• 1 −

ω2

p

ω2

= 1. This model is anisotropic and dispersive

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 15 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Uniaxial Plasma Model: Slowness Curves

ω < ωp

ky kz −1 1 −1 1

Vg Vph Direction ez: only forward waves ω > ωp

ky kz −1 1 −1 1

Vg Vph Direction ez: only forward waves

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 16 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Uniaxial Plasma Model: Slowness Curves

ω < ωp

ky kz −1 1 −1 1

Vg Vph Direction ez: only forward waves ω > ωp

ky kz −1 1 −1 1

Vg Vph Direction ez: only forward waves Use B´ erenger’s PML in this direction

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 16 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Uniaxial Plasma Model: Slowness Curves

ω < ωp

ky kz −1 1 −1 1

Vg Vph Direction ey: only backward waves ω > ωp

ky kz −1 1 −1 1

Vg Vph Direction ey: only forward waves

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 17 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Construction of Stable PMLs

• E. B´

ecache, P. Joly, M. Kachanovska, V. Vinoles (hal-01082445v2): Drude Model Plasma ωm ωe k ωp ky

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 18 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Construction of Stable PMLs

• E. B´

ecache, P. Joly, M. Kachanovska, V. Vinoles (hal-01082445v2): Drude Model Plasma ωm ωe k ωp ky Hence, in the direction (ky, 0) we construct the perfectly matched layer using a modified change

• f variables:

y → y + 1 iωψ(ω)

y

• σ(y′)dy′,

where ψ(ω) = 1 −

ω2

p

ω2 .

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 18 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 2D

Stability of the Modified PML for Maxwell Equations in Uniaxial Plasma in 2D

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 19 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Uniaxial Cold Plasma Model in 3D

∂tE − curl B + jpe3 = 0, ∂tB + curl E = 0, ∂tjp = ω2

pEz.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 20 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Uniaxial Cold Plasma Model in 3D

∂tE − curl B + jpe3 = 0, ∂tB + curl E = 0, ∂tjp = ω2

pEz.

In the frequency domain (∂t → iω, ∂x → −ikx)  (k2 − ω2)I − kkT + ω2

p

  1     E = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 20 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Uniaxial Cold Plasma Model in 3D

∂tE − curl B + jpe3 = 0, ∂tB + curl E = 0, ∂tjp = ω2

pEz.

In the frequency domain (∂t → iω, ∂x → −ikx)  (k2 − ω2)I − kkT + ω2

p

  1     E = 0. Dispersion Relation F(ω, k) =

• ω2 − k2

ω4 − ω2(ω2

p + k2) + k2 z ω2 p

• =
• ω2 − k2

F2D

• ω, (k2

x + k2 y )

1 2 , kz

• .
• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 20 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Slowness Surfaces

ω < ωp

• k2

x + k2 y

kz −1 1 −1 1

Vph Vg Vph Vg Direction ey: both forward and backward waves Direction ez: only forward waves ω > ωp

• k2

x + k2 y

kz −1 1 −1 1

Vph Vg Vg Vph Only forward waves

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 21 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Dispersion relation: F(ω, k) =

• ω2 − k2

F2D

• ω, (k2

x + k2 y )

1 2 , kz

• .
• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 22 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Dispersion relation: F(ω, k) =

• ω2 − k2

F2D

• ω, (k2

x + k2 y )

1 2 , kz

• .

Construct two system of equations one of which will have a dispersion relation F1(ω, k) = ω2 − k2 and another one F2(ω, k) = F2D

• ω, (k2

x + k2 y )

1 2 , kz

• ,

so that F(ω, k) = F1(ω, k)F2(ω, k).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 22 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Dispersion relation: F(ω, k) =

• ω2 − k2

F2D

• ω, (k2

x + k2 y )

1 2 , kz

• .

Construct two system of equations one of which will have a dispersion relation F1(ω, k) = ω2 − k2 and another one F2(ω, k) = F2D

• ω, (k2

x + k2 y )

1 2 , kz

• ,

so that F(ω, k) = F1(ω, k)F2(ω, k). B0 = (0, 0, B0) x y z Maxwell Split PML

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 22 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Dispersion relation: F(ω, k) =

• ω2 − k2

F2D

• ω, (k2

x + k2 y )

1 2 , kz

• .

Construct two system of equations one of which will have a dispersion relation F1(ω, k) = ω2 − k2 and another one F2(ω, k) = F2D

• ω, (k2

x + k2 y )

1 2 , kz

• ,

so that F(ω, k) = F1(ω, k)F2(ω, k). B0 = (0, 0, B0) x y z Maxwell Split PML Inside the PML: the system with the dispersion relation ω2 = k2: B´ erenger PML the system with F2D(ω, (k2

x + k2 y )

1 2 , kz) = 0:

the new PML (new change of variables x, y and the B´ erenger PML in the direction ˆ z)

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 22 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Solutions F1(ω, k) = ω2 − k2

Recall:  (k2 − ω2)I − kkT + ω2

p

  1     E = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 23 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Solutions F1(ω, k) = ω2 − k2

Recall:  (k2 − ω2)I − kkT + ω2

p

  1     E = 0. Inserting the solution ω = ±|k| into the above: kxkT E = 0 kykT E = 0 kzkT E − ω2

pEz = 0

   = ⇒

• kT E = 0,

Ez = 0. I.e. kxEx + kyEy = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 23 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Solutions F1(ω, k) = ω2 − k2

Recall:  (k2 − ω2)I − kkT + ω2

p

  1     E = 0. Inserting the solution ω = ±|k| into the above: kxkT E = 0 kykT E = 0 kzkT E − ω2

pEz = 0

   = ⇒

• kT E = 0,

Ez = 0. I.e. kxEx + kyEy = 0. Conclusion Solutions corresponding to modes ω = ±|k| are 2D divergence-free, i.e. ∂xEx + ∂yEy = 0, and also Ez = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 23 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Solutions F2(ω, k) = F2D(ω, (k2

x + k2 y )

1 2 , kz).

Recall:  (k2 − ω2)I − kkT + ω2

p

  1     E = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 24 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Solutions F2(ω, k) = F2D(ω, (k2

x + k2 y )

1 2 , kz).

Recall:  (k2 − ω2)I − kkT + ω2

p

  1     E = 0. A similar procedure gives, with α =

• (k2 + ω2

p)2 − 4k2 z ω2 p: k2−ω2

p±α

2

Ex = kxkT E

k2−ω2

p±α

2

Ey = kykT E    = ⇒ (kyEx − kxEy)

k2−ω2

p±α

2

= 0. Since k2 − ω2

p ± α = 0 ⇐

⇒ kx = ky = 0 or ωp = 0. it holds kyEx − kxEy = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 24 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Solutions F2(ω, k) = F2D(ω, (k2

x + k2 y )

1 2 , kz).

Recall:  (k2 − ω2)I − kkT + ω2

p

  1     E = 0. A similar procedure gives, with α =

• (k2 + ω2

p)2 − 4k2 z ω2 p: k2−ω2

p±α

2

Ex = kxkT E

k2−ω2

p±α

2

Ey = kykT E    = ⇒ (kyEx − kxEy)

k2−ω2

p±α

2

= 0. Since k2 − ω2

p ± α = 0 ⇐

⇒ kx = ky = 0 or ωp = 0. it holds kyEx − kxEy = 0. Conclusion Solutions corresponding to modes F2(ω, k) = 0 are 2D curl-free, i.e. ∂yEx − ∂xEy = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 24 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Splitting

B0 = (0, 0, B0) x y z L2 = L2

• R3

, H1

⊥ = {φ ∈ L2 : ∂xφ, ∂yφ ∈ L2} .

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 25 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Splitting

B0 = (0, 0, B0) x y z L2 = L2

• R3

, H1

⊥ = {φ ∈ L2 : ∂xφ, ∂yφ ∈ L2} .

Then L2

2 = curl⊥ H1 x,y ⊥

+ grad⊥H1

x,y,

div⊥ curl⊥ H1

x,y = 0,

curl⊥ grad⊥H1

x,y = 0.

We split E⊥ = Ex Ey

• = Ec + Eg,

Ec ∈ curl⊥ H1

x,y, Eg ∈ grad⊥H1 x,y.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 25 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Splitting

B0 = (0, 0, B0) x y z L2 = L2

• R3

, H1

⊥ = {φ ∈ L2 : ∂xφ, ∂yφ ∈ L2} .

Then L2

2 = curl⊥ H1 x,y ⊥

+ grad⊥H1

x,y,

div⊥ curl⊥ H1

x,y = 0,

curl⊥ grad⊥H1

x,y = 0.

We split E⊥ = Ex Ey

• = Ec + Eg,

Ec ∈ curl⊥ H1

x,y, Eg ∈ grad⊥H1 x,y.

Similarly, B⊥ = Bc + Bg.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 25 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Splitting

B0 = (0, 0, B0) x y z L2 = L2

• R3

, H1

⊥ = {φ ∈ L2 : ∂xφ, ∂yφ ∈ L2} .

Then L2

2 = curl⊥ H1 x,y ⊥

+ grad⊥H1

x,y,

div⊥ curl⊥ H1

x,y = 0,

curl⊥ grad⊥H1

x,y = 0.

We split E⊥ = Ex Ey

• = Ec + Eg,

Ec ∈ curl⊥ H1

x,y, Eg ∈ grad⊥H1 x,y.

Similarly, B⊥ = Bc + Bg. NB: we do not split Ez, Bz, jp

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 25 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) TM mode (Eg, Bc, Ez, jp)

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) TM mode (Eg, Bc, Ez, jp)

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0,

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0,

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0,

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0,

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0,

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Construction of the Split System

The original Maxwell system: ∂tE⊥ − curl⊥ Bz −

• e3 × ∂z

B⊥

• ⊥

= 0, ∂tEz − curl B⊥ + jp = 0, ∂tjp = ω2

pEz,

∂tB⊥ + curl⊥ Ez +

• e3 × ∂z

E⊥

• ⊥

= 0, ∂tBz + curl E⊥ = 0. TE mode (Ec, Bg, Bz) ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. TM mode (Eg, Bc, Ez, jp) ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

Equivalence in R3 if curl⊥ Eg = 0, curl⊥ Bg = 0 (can be ensured by a proper choice of initial conditions)

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 26 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. The dispersion relation: FTE (ω, k) = (k2

z − ω2)(k2 − ω2)

= (k2

z − ω2)F1(ω, k).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. The dispersion relation: FTE (ω, k) = (k2

z − ω2)(k2 − ω2)

= (k2

z − ω2)F1(ω, k).

Only forward propagating modes (hence B´ erenger’s PML)

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. The dispersion relation: FTE (ω, k) = (k2

z − ω2)(k2 − ω2)

= (k2

z − ω2)F1(ω, k).

Only forward propagating modes (hence B´ erenger’s PML) TM Mode ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. The dispersion relation: FTE (ω, k) = (k2

z − ω2)(k2 − ω2)

= (k2

z − ω2)F1(ω, k).

Only forward propagating modes (hence B´ erenger’s PML) TM Mode ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

The dispersion relation: FTM(ω, k) = (k2

z − ω2)

 k2

z + k2 2 + k2 1

1 −

ω2

p

ω2

− ω2   = (k2

z − ω2)F2(ω, k).

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. The dispersion relation: FTE (ω, k) = (k2

z − ω2)(k2 − ω2)

= (k2

z − ω2)F1(ω, k).

Only forward propagating modes (hence B´ erenger’s PML) TM Mode ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

The dispersion relation: FTM(ω, k) = (k2

z − ω2)

 k2

z + k2 2 + k2 1

1 −

ω2

p

ω2

− ω2   = (k2

z − ω2)F2(ω, k).

For ω ≥ ωp all waves propagate forward For ω < ωp waves propagate backwards in directions ex, ey and forward in ez

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Split System

TE Mode ∂tEc − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂tBg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂tBz + curl Ec = 0. The dispersion relation: FTE (ω, k) = (k2

z − ω2)(k2 − ω2)

= (k2

z − ω2)F1(ω, k).

Only forward propagating modes (hence B´ erenger’s PML) TM Mode ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

The dispersion relation: FTM(ω, k) = (k2

z − ω2)

 k2

z + k2 2 + k2 1

1 −

ω2

p

ω2

− ω2   = (k2

z − ω2)F2(ω, k).

For ω ≥ ωp all waves propagate forward For ω < ωp waves propagate backwards in directions ex, ey and forward in ez 4 extra unknowns = ⇒ extra modes

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 27 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

TM Mode

TM Mode ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 28 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

TM Mode

TM Mode ∂tEg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂tBc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂tEz − curl Bc + jp = 0, ∂tjp = ω2

pEz.

PML Change of Variables (ψ(ω) = 1 −

ω2

p

ω2 ):

x → x + 1 iω

• 1 −

ω2

p

ω2

• x
• σ(x′)dx′,

y → y + 1 iω

• 1 −

ω2

p

ω2

• y
• σ(y′)dy′,

z → z + 1 iω

z

• σ(z′)dz′.
• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 28 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Discretization

Structure similar to that of Maxwell equations = ⇒ Yee scheme.

Ex Ey Ez By Bz Bx

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 29 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Discretization

Structure similar to that of Maxwell equations = ⇒ Yee scheme.

Ex Ey Ez By Bz Bx

∂t Ec − curl⊥ Bz −

• e3 × ∂z

Bg

• ⊥

= 0, ∂t Bg +

• e3 × ∂z

Ec

• ⊥

= 0, ∂t Bz + curl Ec = 0. Ecx Ecy Bgy Bz Bgx ∂t Eg −

• e3 × ∂z

Bc

• ⊥

= 0, ∂t Bc + curl⊥ Ez +

• e3 × ∂z

Eg

• ⊥

= 0, ∂t Ez − curl Bc + jp = 0, ∂t jp = ω2 pEz . Egx Egy Ez Bcy Bcx

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 29 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Coupling on the Discrete Level

Example of the coupling for Bz: Ex Ex Ey Ey Ecx Egx Ecy Egy Ecy Egy

Ex = Ecx + Egx ∂x Egy − ∂y Egx = 0

Similarly for Ez.

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Coupling on the Discrete Level

Example of the coupling for Bz: Ex Ex Ey Ey Ecx Egx Ecy Egy Ecy Egy

B n+ 1 2 z = B n− 1 2 z − h

• ∂x En

y − ∂y En x

• Ex = Ecx + Egx

∂x Egy − ∂y Egx = 0

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Coupling on the Discrete Level

Example of the coupling for Bz: Ex Ex Ey Ey Ecx Egx Ecy Egy Ecy Egy

B n+ 1 2 z = B n− 1 2 z − h

• ∂x En

y − ∂y En x

• B

n+ 1 2 z = B n− 1 2 z − h

• ∂x En

cy − ∂y En cx

• Ex = Ecx + Egx

∂x Egy − ∂y Egx = 0

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Coupling on the Discrete Level

Example of the coupling for Bz: Ex Ex Ey Ey Ecx Egx Ecy Egy Ecy Egy

B n+ 1 2 z = B n− 1 2 z − h

• ∂x En

y − ∂y En x

• B

n+ 1 2 z = B n− 1 2 z − h

• ∂x En

cy − ∂y En cx

• B

n+ 1 2 z = B n− 1 2 z − h

• ∂x (En

cy + En gy ) − ∂y (En cx + En gx )

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Coupling on the Discrete Level

Example of the coupling for Bz: Ex Ex Ey Ey Ecx Egx Ecy Egy Ecy Egy

B n+ 1 2 z = B n− 1 2 z − h

• ∂x En

y − ∂y En x

• B

n+ 1 2 z = B n− 1 2 z − h

• ∂x En

cy − ∂y En cx

• B

n+ 1 2 z = B n− 1 2 z − h

• ∂x (En

cy + En gy ) − ∂y (En cx + En gx )

• Ex = Ecx + Egx

∂x Egy − ∂y Egx = 0

Similarly for Ez.

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 30 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Numerical Experiment

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 31 / 32

PMLs for Plasmas Uniaxial Cold Plasma Model in 3D

Numerical Experiment

1

the source is in the center of the box;

2

we plot Ez for a cross-section z = const and Ex for a cross-section x = const;

3

conventional PML: applied to the splitted system by setting ψ(ω) = 1. Solution

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 31 / 32

Conclusions

Conclusions and Open Questions

1

stable PML for the 2D Maxwell equations in plasmas if ωc → ∞

2

stable PML for the 3D Maxwell equations in plasmas if ωc → ∞:

concept and idea numerical implementation (Yee scheme) stable variational formulation of the coupling ?

3

PMLs for arbitrary external magnetic field?

• E. B´

ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 32 / 32