The Riemann problem for a full-wave Maxwell system modeling - - PowerPoint PPT Presentation

The Riemann problem for a full-wave Maxwell system modeling electromagnetic propagation in a nonlinear Kerr medium

Denise Aregba-Driollet

• Univ. Bordeaux, IMB.

Kerr’s model

Maxwell’s equations: ∂tD − curlH = ∂tB + curlE = divD = divB = 0 Constitutive laws: B = µ0H D = ǫ0(1 + ǫr|E|2)E

Kerr model

∂tD − curlH = ∂tH + µ−1

0 curlP(D)

= where P = Q−1 and Q(E) = ǫ0(1 + ǫr|E|2)E. divD = divH = 0. Mathematical entropy : electromagnetic energy. Hyperbolic symmetrizable system of conservation laws. Notation: E = P(D).

Properties of the Kerr model

◮ Eigenvalues in a given direction ω ∈ R3, |ω| = 1:

λ1 ≤ λ2 < λ3 = λ4 = 0 < λ5 = −λ2 ≤ λ6 = −λ1 with λ2

1 =

c2 1 + ǫr|E|2 , λ2

2 = c2 1 + ǫr(|E|2 + 2(E · ω)2)

(1 + ǫr|E|2)(1 + 3ǫr|E|2). Moreover λ2

1 = λ2 2 if and only if D × ω = 0. ◮ The characteristic fields 1, 3, 4, 6 are linearly degenerate. ◮ In the open domain {D ∈ R3; D × ω = 0} the characteristic

fields 2, 5 are genuinely nonlinear.

2D reduced models with spatial variable x = (x1, x2)

Transverse Magnetic (TM): H = (0, 0, h), D = (D1, D2, 0). Transverse Electric (TE): D = (0, 0, d), H = (H1, H2, 0). In each case : 3 × 3 strictly hyperbolic system of conservation laws with 0 as simple eigenvalue. Eigenvalues 1 and 6 of the 6 × 6 system are lost.

A 1D reduced model with spatial variable x = x1

D = (O, d, 0), H = (0, 0, h). divD = divH = 0 always. Kerr system reads as a p-system: ∂td + ∂xh = 0, ∂th + µ−1

0 ∂xp(d) = 0.

• 2e+09
• 1.5e+09
• 1e+09
• 5e+08

5e+08 1e+09 1.5e+09 2e+09

• 0.1
• 0.05

0.05 0.1 p(d) d

A relaxation system for Kerr model: Kerr-Debye system

         ∂tD − curlH = ∂tH + µ−1

0 curlE

= 0, E = D ǫ0(1 + χ) ∂tχ = −1 τ

• χ − ǫr|E|2

In situations of physical interest τ is very small. Hyperbolic, partially dissipative system, relaxation of Kerr model in the sense of Chen-Levermore-Liu (CPAM 1994).

R.W. Ziolkowski. IEEE Transactions on Antennas and Propagation 45(3):375-391, 1997.

• P. Huynh. PhD thesis, 1999.

Goal and motivations

◮ Relaxation and existence for strong solutions: see

Carbou-Hanouzet (JHDE 2009), Kanso (PhD thesis 2012).

◮ Study of shocks and related Kerr-Debye shock profiles:

AD-Hanouzet, CMS 2011

◮ Numerical approximation of Kerr system by a Kerr-Debye

relaxation finite volumes scheme: AD-Berthon 2009 in 1D, Kanso PhD thesis in 2D.

◮ Here, we study the Riemann problem for the 6 × 6 system and

relate its solutions with those of the reduced models in order to

◮ Perform numerical approximation by Godunov scheme ◮ Understand better the weak solutions

See also A. de la Bourdonnaye JCP 2000, for Godunov type schemes for reduced Kerr systems.

Notation: u = (D, H). We fix ω ∈ R3, |ω| = 1, ul ∈ R6, ur ∈ R6, u(x, 0) = ul if x · ω < 0, ur if x · ω > 0. We look for a solution under the form u(x, t) = u(x · ω, t). One-dimensional 6 × 6 Riemann problem with variable y = x · ω.

Study of the simple waves: contact discontinuities

If (D, H) is a stationary contact discontinuity such that divD = 0 and divH = 0, then it is a constant function. Others contact discontinuities: related to the eigenvalues 1 and 6. Rotating modes such that |D| and divD, divH are constant.

Dl ω Dr

Study of the simple waves: shock waves and rarefactions

Eigenvalues 2 and 5.

◮ If Dl × ω = 0 one can define a 2-rarefaction curve, a Lax

2-shock curve and a Liu 2-shock curve.

◮ If Dl × ω = 0, one has rarefactions and semi-contact

discontinuities. Lax conditions ensure that the Riemann fan can be constructed for the 6 × 6 system. Liu’s condition (J. Math. Anal. Appl., 1976): if ul is a left state which the Hugoniot set H(ul) is a union of curves and if ur ∈ H(ul). σ(ur, ul) ≤ σ(u, ul), ∀u ∈ H(ul), u between ul and ur .

D component for a 2-wave when Dl × ω = 0

−ω×(ω× Dl) ω Dr D* Dl

Liu 2-shock

−ω×(ω× Dl) ω Dl Dr

Lax 2-shock

−ω×(ω× Dl) ω D* Dl Dr

2-rarefaction

Solution of the 6 × 6 Riemann problem

Initial data: u0(x) = ul if x · ω < 0, ur if x · ω > 0. We show that for |ur − ul| small enough, there exists a unique solution of the form

ul ur u1 u2 u* u** t y 1−cd 6−cd 2−shock or rarefaction 5−shock or rarefaction stationary cd

If divD0 = divH0 = 0

◮ The solution exists without smallness condition. ◮ u∗ = u∗∗: no stationary contact discontinuity. ◮ If moreover Hr − Hl − ω × (σrDr − σlDl) = 0

D∗ × ω = 0. 1-cd and 2-shock (resp 6-cd and 5-shock) merge.

ul ur u* y t

The case of the 2 × 2 reduced model

p-system with p convex-concave. Two solutions can be constructed:

◮ Weak solution as a particular case of the 6 × 6 system. ◮ ”Liu’s solution”, see also Wendroff, J. Math. Anal. Appl.

1972.

d component at fixed time

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

• 6e-07
• 5e-07
• 4e-07
• 3e-07
• 2e-07
• 1e-07

1e-07 2e-07 3e-07 4e-07 Liu 2x2 solution: d 6x6 solution: d

h component at fixed time

• 1.5e+07
• 1e+07
• 5e+06

5e+06 1e+07 1.5e+07 2e+07

• 6e-07
• 5e-07
• 4e-07
• 3e-07
• 2e-07
• 1e-07

1e-07 2e-07 3e-07 4e-07 Liu 2x2 solution: h 6x6 solution: h

d component in another case

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

• 4e-07
• 3e-07
• 2e-07
• 1e-07

1e-07 2e-07 3e-07 4e-07 Liu 2x2 solution: d 6x6 solution: d

The case of the 2 × 2 reduced model

Entropy: electromagnetic energy η(d, h) = E(d) + 1 2µ0h2, E(d) = ǫ0 2

• e2 + 3ǫr

2 e4

• with e = p(d).

Entropy flux: Poynting vector Q(d, h) = eh. Liu’s shocks satisfy the following entropy dissipation inequality: [Q(d, h)] − σ[η(d, h)] = −ǫ0ǫr 4 σ[e]2 [e2] ≤ 0.

The case of the 2 × 2 reduced model

◮ Both solutions are entropic. ◮ Liu’s solution is in general more dissipative than the 6 × 6

solution because

• 1. the entropy dissipation rate is shown to increase with |[d]|,

which is larger for Liu’s shocks than for 6 × 6 Lax shocks,

• 2. for contact discontinuities, entropy is conserved.

◮ Numerically, the solutions of the relaxation Kerr-Debye system

converge towards Liu’s solutions.

Conclusion and perspectives

◮ For |ur − ul| small enough we have constructed the solution of

Riemann problem for the 3D Kerr system as a composition of simple waves.

◮ In the divergence free case, the solution exists for any

Riemann data.

◮ For the 2 × 2 reduced model, we have two entropy solutions. ◮ Numerical application: Godunov scheme, to be compared with

already existing relaxation Kerr-Debye scheme (partially done).