Theoretical and numerical analysis of local dispersion models coupled to a discontinuous Galerkin time-domain method for Maxwell's equations

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Theoretical and numerical analysis of local dispersion

models coupled to a discontinuous Galerkin time-domain

method for Maxwell’s equations

Jonathan Viquerat, Maciej Klemm, Stéphane Lanteri, Claire Scheid

To cite this version:

Jonathan Viquerat, Maciej Klemm, Stéphane Lanteri, Claire Scheid. Theoretical and numerical

anal-ysis of local dispersion models coupled to a discontinuous Galerkin time-domain method for Maxwell’s

equations. [Research Report] RR-8298, INRIA. 2013, pp.79. �hal-00819758v2�

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ISRN INRIA/RR--8298--FR+ENG

RESEARCH

REPORT

N° 8298

numerical analysis of

local dispersion models

coupled to a

discontinuous Galerkin

time-domain method for

Maxwell’s equations

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RESEARCH CENTRE

discontinuous Galerkin time-domain method

for Maxwell’s equations

Jonathan Viquerat, Maciej Klemm

∗

_{, Stéphane Lanteri,}

Claire Scheid

†

Project-Team Nachos

Research Report n° 8298 — 3 Mai 2013 —76pages

Abstract: This report focuses on a centered-fluxes discontinuous Galerkin method coupled to a second-order Leap-Frog time scheme for the propagation of electromagnetic waves in dispersive media. After a presentation of the physical phenomenon and the classical disper-sion models (particularly the Drude one), a generalized dispersive model is introduced. An

a priori stability and convergence study is lead for the Drude model, as well as in the

gen-eralized dispersive case. Eventually, numerical results are presented for various test-cases, highlighting the interest of a proper description of the dispersion phenomenon in metals at the nanoscale.

Key-words: Discontinuous Galerkin method, Maxwell’s equations, numerical electromag-netism, dispersive media, nanophotonics, Drude model

∗_{University of Bristol}

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Theoretical and numerical analysis of local dispersion

models coupled to a discontinuous Galerkin time-domain

method for Maxwell’s equations

Résumé : Ce rapport présente une méthode de Galerkin discontinue à flux centrés cou-plée à un schéma d’avancée en temps de type Leap-Frog d’ordre deux pour la propaga-tion des ondes électromagnétiques dans les milieux dispersifs. Après une présentapropaga-tion du phénomène physique ainsi que des modèles de dispersion les plus classiques (notamment celui de Drude), un modèle de dispersion généralisé est introduit. Une étude de stabilité et de convergence a priori est conduite dans le cas du modèle de Drude, ainsi que dans le cas généralisé. Enfin, des résultats numériques sont présentés pour différents cas-tests, met-tant en lumière l’intérêt d’une bonne description des phénomènes de dispersion des métaux à l’échelle nanoscopique.

Mots-clés : Méthode de Galerkin discontinue, équations de Maxwell, électromagnétisme numérique, milieux dispersifs, nanophotonique, modèle de Drude.

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List of Figures

1 Real and imaginary parts of the silver relative permittivity predicted by the Drude model compared to experimental data . . . 12 2 Real and imaginary parts of the silver relative permittivity predicted by the

Drude-Lorentz model compared to experimental data . . . 14 3 Real and imaginary parts of the silver relative permittivity predicted by the

2-SOGP model compared to experimental data . . . 17 4 Update scheme in the genrally dispersive case . . . 29 5 Exact and calculatedExfields with aP1approximation for the artificial case . . 57 6 L2_{error for different orders of approximation, for the artificial case} _{. . . 58} 7 Physical situation : gold nanosphere illuminated with a plane wave . . . 59 8 Comparison of DG and Mie solutions for the near-field enhancement of a gold

nanosphere . . . 60 9 Mie and DG 1D plot of the electric field modulus across the dispersive gold

nanosphere . . . 61 10 Comparison of two surfacic meshes for the gold nanosphere . . . 62 11 Physical situation : silver slab illuminated with a plane wave . . . 63 12 Reflection coefficient of a silver slab described by several dispersion models . . 65 13 Computational time and memory allocation for the generalized dispersive

for-mulation for various models . . . 66 14 Comparison of the fitting of imaginary part of the silver permittivity by 2-SOGP

and 4-SOGP and its effect on the prediction of the reflectance spectrum . . . . 66 15 Real and imaginary parts of the silver and gold relative permittivity predicted

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List of Tables

1 Drude parameters for silver . . . 13 2 Drude-Lorentz parameters for silver . . . 14 3 Parameters of the generally dispersive permittivity function for the Drude,

Drude-Lorentz and Debye models . . . 15 4 2-SOGP parameters for silver . . . 17 5 Convergence orders for the artificial case withP₁andP₂approximations . . . . 57 6 Parameters set for the gold nanosphere case . . . 58 7 L1 _{errors for various meshes and orders of approximation in the case of the}

gold nanosphere nearfield enhancement . . . 59 8 L1 _{errors, computational times and allocated memory for various dispersive}

model in the silver slab case . . . 64 9 Coefficients of various dispersive models for silver . . . 69 10 Coefficients of various dispersive models for gold . . . 70

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1 Photonics and dispersive media

The increasing need in electronic signals processing during the last century came along with an advanced miniaturization of all the associated components. The revolution that followed the advent of the transistor deeply modified the common use of information. In the seven-ties, the development of the optical fiber led the way to fastest and largest data exchanges by exploiting light flow instead of electron flow. A precise control of the electromagnetic waves (confinment, transmitted frequencies, propagation direction) using wavelength-size systems would permit to replace the electronic-based management of the information. For many fields (e.g. optics, informatics, medicine), the benefits withdrawned from such an outcome would be invaluable.

The particular properties required by such devices are artificial, i.e. they are not dis-played by any natural material. The macroscopic behavior of these "metamaterials" usually results from their nanoscopic structure, which induces particular interactions with propa-gating waves. Under this vocable are included the negative-index materials [VBSH06], as well as photonic cristals [JJ07], for example.

Then, the fast-growing metamaterials development is due to their innovative features at the macroscopic scale. However, one must not forget that these properties rely on those of "regular" materials (dielectrics, metals, etc.) assembled at the nanoscopic scale. Hence, poorly modelized attributes for the latter would result in imprecisely predicted properties for the resulting metamaterial. Among others, physical dispersion has a great impact on the material/electromagnetic wave interaction; therefore, attention has been called on the various ways of modeling physical dispersion, especially in metals.

1.1 Physical dispersion

1.1.1 Definition

Dispersion is a common phenomenon to all kinds of waves traveling through a material medium : it results from the way the latter reacts to the presence of the wave, therefore affecting its propagation. For a polychromatic wave, it often happens that all the frequencies do not travel at the same speed through the medium : this phenomenon is called dispersion. In order to characterize it, one generaly tries to express a dispersion relation that bounds the angular frequencyωto the wave numberk.

1.1.2 Dispersion relation

Since it links a temporal parameterωto a spatial onek, this relationω= f (k)describes the allowed modes inside the considered domain. It is often written as :

ω(k) = v(k)k, (1)

wherev(k)is the speed of the monochromatic wave of wavenumberk. This relation is often explicit in the most simple cases, but it can become implicit in more complicated situations. Let us now consider a polychromatic wave that can be expanded into a sum of monochro-matic modes. In the case of a non-dispersive medium,v(k)would remain constant for every value ofk, and all the frequencies would travel at the same speed. On the contrary, if the

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10 Lanteri, Scheid & Viquerat

medium is dispersive, (1) indicates that each frequency travels at a different speed. There-fore, a signal consisting of frequencies that are initially spatially localized will sprawl during its propagation. The description of this phenomenon calls for the definition of two different speeds :

• The speed of the wavefront, usually called phase velocity :vφ=

ω(k) k , • The speed of the wave envelope, called group velocity : vg=

∂ω ∂k.

These two velocities are equal in the case of a non-dispersive medium. Otherwise, the group velocity is the physical speed of the problem, given it is linked to the speed at which the energy moves through the medium. On the other hand, the phase velocity is more artificial, since it represents the speed at which a point of constant phase moves through the medium. One could imagine situations for which the phase velocity exceeds the speed of lightc, without violating any fundamental principle of physics.

We now restrain our study to the case of propagating electromagnetic waves (EMW) through dispersive metals, which is at the heart of to the numerical study that will follow. 1.1.3 Physical origin of the dispersion in metals for EMW

In the presence of a constant electric field, the electrons of a metal are subjected to a Coulomb force which brings them, in a given characteristic timeτc, to an equilibrium posi-tion. This leads to a general electric polarization of the metal, which is usually expressed with the polarization vector P. The latter constitutes an additional term to the electric

dis-placement field D : D= ε0E+P. Moreover, P can be related to E in homogeneous isotropic media through its susceptibilityχsuch that P= χE. One should now grasp the importance of taking the dispersion effects into account when P cannot be neglected, since it has a great influence on the permittivityεof the considered medium, and hence on its optical index.

If one is to consider a variable electric field of given angular frequencyω, the frequency dependence of P can be intuitively understood : for low enough frequencies, the electrons relaxation time τ is negligible compared to 1

ω. Therefore, the electrons dispose of a suf-ficient amount of time to adapt to the variations of the electric field. However, at higher frequencies, the field varies significantly during the time τ required by the electrons to reach a stable state. Then, the higher the frequency, the shorter the distance traveled by the electrons from their steady state equilibrium, and the lower the polarization. This ex-plains the observed transparency of the metals for very high frequencies EMW. Moreover, one can now easily picture that the electrons contributing to the polarization mainly belong to the conduction band, since they are less bound to the nuclei, and therefore more movable. It is now possible to introduce the most common modelisations of the disperion phe-nomenon in metals. Among others, the well-known Drude and Drude-Lorentz models will be presented.

1.2 Drude model

The discovery of the electron in 1897 by Thomson was followed, three years later, by the Drude theory based on the kinetic theory of gases [Dru00]. This particularly simple the-ory successfully accounts for the optical and thermical properties of some metals. In this Inria

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model, the metal is considered as a static lattice of positive ions immersed in a free elec-trons gas. Those elecelec-trons are considered to be the valence elecelec-trons of each metallic atom, that got delocalized when put into contact with the potential produced by the rest of the lattice atoms. They are often called conduction electrons, and wander freely in the

conduc-tion band, the core electrons being considered to be strongly bound to the ion core. As a

consequence, the Drude model is often described as an intraband model. Moreover, several hypothesis are made :

• The electrons description is non-relativistic;

• The only considered interactions are the electron/wave and the electron/ion ones;

• The electron/ion collisions are instantaneous and random events, and their probability of happening during a dtamount of time is equal to dt

τf

1_;

• After an electron/ion collision, the new velocity and direction of the electron are inde-pendant of those before the collision.

Under these hypotheses, the frequency dependence of the medium permittivity can be de-duced from the equations of motion.

1.2.1 Relative permittivity

In the absence of relativistic effects (as the Laplace force), the velocity equation of an elec-tron can be written as follows :

∂v

∂t + γdv=− e me

E(t), (2)

wheremerepresents the electron mass,ethe electronic charge, andγd a coefficient linked to the electron/ion collisions. One could notice that[γd] = T−1, and therefore γd matches the definition of the inverse of the mean free pathτf.

For an harmonic field (i.e. E=E0e−iωt), the electron velocity will be of the form v(t) = v0e−iωt. Then, (2) leads to :

v0=−

eE0

me(γd− iω)

. (3)

Combining the current density2_J₌_−n

eev with (3) and Ohm’s law3J= σdE, one obtains :

σd(ω) = nee2 me 1 γd− iω . (4)

The relative permittivity is linked to its conductivity with the relationεr,d(ω) = ε∞+ _ωε0iσd. Here,ε∞represents the core electrons contribution, which is equivalent to approximate the positive ions lattice as a continuous medium of permittivityε∞. For the metals successfully described by the Drude model, ε∞ is close to 1 (taking ε∞ different from 1 is sometimes refered to as the Drude-Sommerfeld model). Eventually, one gets :

1_τ

f is the electron mean free path. 2_n

estands for the electronic density. 3_σ

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12 Lanteri, Scheid & Viquerat εr,d(ω) = ε∞− ω2 d ω2_{+ iωγ} d , (5) where ωd = q nee2

meε0 is the plasma frequency of the electrons. The latter is linked to a typical relaxation time of slightly perturbed plasma electrons around their thermodynamic equilibrium state. By separating the real and imaginary parts of (5) :

εr,d = ε1+ iε2 = ε∞− ω2_d γ2 d+ ω2 + i γdω 2 d ω(γ2 d+ ω2) ,

and by considering ω _{≪ γ}d (which remains particularly accurate in the visible frequency field for the metals described by the Drude model), the above expressions can be simplified as : ε1(ω)≃ ε∞− ω2_d ω2 andε2(ω)≃ ω2_dγd ω3 ,

which enlights the role of the parameterγdin the dissipation induced by the Drude model4. The real and imaginary parts of the Drude permittivity function of the silver are plotted in figure 1, along with experimental curves from Johnson and Christy [JC72]. The used parameters can be found in table1.

2 4 6 8 −40 −20 0 ω(PHz) ε1 (ω ) Drude Johnson & Christy

(a) Real part

2 4 6 8 0 1 2 3 4 ω(PHz) ε1 (ω ) Drude Johnson & Christy

(b) Imaginary part

Figure 1 | Real and imaginary parts of the silver relative permittivity predicted by the Drude model compared to experimental data from Johnson & Christy.

One should notice that, if the real part fits the Drude model prediction, the experimental imaginary parts show features that are not predicted by the model. Those latter root in

interband phenomenons, i.e. core electrons contributions that jump to the conduction band.

These features can be better fitted with the Drude-Lorentz model, which is discussed in the following section.

4_Indeed,_γ

drepresents the friction experienced by the electrons.

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Table 1 | Drude parameters for silver in the [300, 1500] THz frequency range. ε∞ ωd γd – GHz GHz 3.7362 1.3871 107 4.5154 104

1.3 Drude-Lorentz model

The initial Drude model describes the behavior of the free electrons contained in the con-duction band. Nevertheless, for certain metals (such as noble ones), electronic transitions between valence and conduction band occur in the visible frequency range, giving them their particular colors. The contributions of these electrons can be taken into account in the permittivity function with additional Lorentz terms. Contrary to the Drude-described electrons, the Lorentz ones are, in some way, "bound" to their ion cores. Therefore, it seems logical to reuse the expression (2), including an additional spring term to it :

∂2_x ∂t2 + γl ∂x ∂t + ω 2 lx=− e me E(t).

Following the same development as for the Drude model, one easily obtains the expression of the Lorentz permittivity :

εr,l(ω) =− ∆εω2 l ω2_{− ω}2 l + iωγl .

The total permittivity can then be written by adding the Drude and Lorentz terms5_:

εr,dl(ω) = ε∞− ω2 d ω2_{+ iωγ} d − ∆εω2 l ω2_{− ω}2 l + iωγl . (6)

As done previously, the previous permittivity can be splitted into a real and an imaginary parts : ε1(ω) = ε∞− ω2 d γ_d2+ ω2− ∆εω2 l ω2− ω2l (ω2_{− ω}2 l) 2 + γ2 lω2 , and ε2(ω) = γdω2d ω(γ2 d+ ω2) + ∆εω 2 lγlω (ω2_{− ω}2 l) 2 + γ2 lω2 .

For some metals such as gold, silver or copper, the addition of Lorentz terms brings a much better fit between experimental and theoretical values. In the case of silver, a Drude-Lorentz model with one Drude-Lorentz pole brings a much better adequation between experiment and theory, especially in the high-frequency range for the imaginary part, as can be seen on figure3. The Drude-Lorentz model parameters used for this fitting are summed up in table 2.

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Table 2 | Drude-Lorentz parameters for silver in the [300, 1500] THz frequency range.

ε∞ ωd γd ∆ε ωl γl – GHz GHz – GHz GHz 2.7311 1.4084 107 6.6786 103 1.6336 8.1286 106 3.6448 106

Nevertheless, it is to be understood that Drude and Drude-Lorentz parameters are fitted from experimental data over a particular range of frequencies. This has several conse-quences, such as(i)the variety of parameter sets that can be found in the literature for the different models (particular attention must be paid to the frequency range of interest);(ii)

in regions of lower regularity of the permittivity function, additional adequatly-chosen poles can be considered to improve the fitting (the next section presents a way of fitting arbitrary permittivity functions). 2 4 6 8 −40 −20 0 ω(PHz) ε1 (ω ) Drude-Lorentz Johnson & Christy

(a) Real part

2 4 6 8 0 1 2 3 4 ω(PHz) ε1 (ω ) Drude-Lorentz Johnson & Christy

(b) Imaginary part

Figure 2 | Real and imaginary parts of the silver relative permittivity predicted by the Drude-Lorentz model compared to experimental data from Johnson & Christy.

1.4 Generalized dispersive model

Given an experimental permittivity function, a Padé approximant would be a convenient analytical coefficient-based function to approach the experimental data. Thus, one could write : εr,g(ω) = n X i=1 αi(jω)i m X k=1 αk(jω)k .

The fundamental theorem of algebra allows to write the previous expression as a decompo-sition of a constant, one zero-order pole (ZOP), a set of first-order generalized poles (FOGP), and a set of second-order generalized poles (SOGP) :

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εr,g(ω) = ε∞− σ jω − X l∈L1 al jω_{− b}l− X l∈L2 cl− jωdl ω2_{− e} l+ jωfl , (7)

where ε∞, σ, al, bl, cl, dl, el, fl are real constants, and L1, L2 are non-overlapping set of in-dices. This general writing allows an important flexibility for two reasons : (i) it unifies most of the common dispersion models in a single formulation (the right coefficient choices are presented in table3for the Drude, Drude-Lorentz and Debye media), and (ii) it permits to fit any experimental data set in a reasonable number of poles (and thus a reasonable number of coefficients), as will be presented later. One should notice that this approach is very similar to the Critical Points (CP) [VLDC11] and the Complex-Conjugate Pole-Residue Pairs (CCPRP) [HDF06]. In facts, these developments are the same in some mathematical sense, since the three of them exploit the fundamental theorem of algebra. The difference between them lies in the way they choose to use it : the CCPRP and the CP allow complex coefficients in their developments, and can therefore write the decomposition of the permit-tivity function in terms of single-order poles only, whereas choosing real coefficients leads to a collection of first-order and second-order poles.

Table 3 | Parameters of the generally dispersive permittivity function for the Drude, Drude-Lorentz and

Debye models. Model al bl cl dl el fl Drude 0 0 ω2 d 0 0 γd Drude-Lorentz 0 0 ∆εω2 l 0 ω 2 l γl Debye ∆εγde −γde 0 0 0 0

The matter of fitting the coefficients of (7) to experimental data remains to be detailed. Various techniques can be used, through which the many existing versions of least square method. Vector fitting techniques [GS99] are also well developed for the CCPRP formu-lation. In our case, a Simulated Annealing (SA) algorithm has been used to determine the coefficients to use given an experimental data set. The original paper describing this method is [KGV83], but the reader can find extensive presentations and discussions by himself over the internet. Notice that the used algorithm has been written by William L. Goffe, and can be found athttp://ideas.repec.org/c/wpa/wuwppr/9406001.html.

A set of coefficients has been calculated for the silver in the same range of frequencies as in the previous sections, using two SOGP. The values of the coefficients are presented in table 4, whereas a plot of the real and imaginary parts of the permittivity function is presented in figure2. This plot is to be compared with the Drude-Lorentz one (see figure3), which can also be seen as a two second-order poles fitting. Although the general expression (7) is not based on a physical model, it seems to display better fittings properties than classical poles such as Lorentz ones. As in the Lorentz case, additional poles can enhance the precision of the fitting.

1.5 Summary

The physical origins of the dispersion for propagating waves have been briefly outlined and common models for EMW dispersion in metals have been presented. A generalized

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disper-16 Lanteri, Scheid & Viquerat

sion models has been proposed that seems to improve the fitting of permittivity function over arbitrary sets of experimental data.

The next part of this report focuses on the Discontinuous Galerkin (DG) method in the time domain, and presents its formulation in the case of the Maxwell’s equations in vacuum. Then, the respective continuous equations and Discontinuous Galerkin Time Domain (DGTD) schemes are deduced for the Drude model, and for the generalized dispersive model.

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Table 4 | 2-SOGP parameters for silver in the [300, 1500] THz frequency range. ε∞ c1 d1 e1 f1 c2 d2 e2 f2 – GHz2 GHz GHz2 GHz GHz2 GHz GHz2 GHz 1.2944 1.8909 1014 2.6584 106 0.0 0.0 5.6165 1013 1.2005 107 4.3932 1013 3.1709 106 2 4 6 8 −40 −20 0 ω(PHz) ε1 (ω ) 2-SOGP Johnson & Christy

(a) Real part

2 4 6 8 0 1 2 3 4 ω(PHz) ε1 (ω ) 2-SOGP Johnson & Christy

(b) Imaginary part

Figure 3 | Real and imaginary parts of the silver relative permittivity predicted by the 2-SOGP model compared to experimental data from Johnson & Christy.

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2 DGTD method for non-dispersive media

This section focuses on the DGTD method for Maxwell equations. LetΩ_{⊂ R}3_{be a bounded} convex domain, and n the outward normal to its boundary ∂Ω. The electric permittivity and permeability of free space are respectively denotedε0andµ0. Then, in free space, the Maxwell equations can be written as follows :

     µ0 ∂H ∂t = −∇ ×E, ε0 ∂E ∂t = ∇ ×H−J, (8)

along with constitutive relations B= µ0H and D= ε0E and metallic boundary conditions. DG methods have been originally introduced in 1973 by Reed and Hill [RH73], and has been widely used since in the computational fluid dynamics field. However, their application to the time-domain Maxwell equations are more recent. DG methods can be seen as classi-cal finite element methods (FEM) for which the global continuity of the approximation has been lifted. This implies that the support of each basis function is restrained to a discretiza-tion cell, which leads to local formuladiscretiza-tions implying no large mass matrix inversion in the process. Afterward, connexion between the cells is restored by the use of a numerical flux to evaluate the boundary integrals. The choice of the numerical flux has a great influence on the mathematical properties of the DG discretization, as energy preservation, for example. Both centered [FLLP05] or upwind [HW02] fluxes can be used.

The discontinuity of the approximation makes room for numerous methodologic improve-ments, such as local approximation orders [Fah09], high parallelization features ([BFLP06], [CCL11]) and the use of non-conformeous [FL10] and hybrid meshes [DLS12], for example. Also, a wide choice of time-integration schemes can be used for the discretization of time derivatives, including Leap-Frog (LF) and Runge-Kutta (RK). As well as local approxima-tion orders can be used, local time-stepping [Pip05] as well as locally implicit formulations [Moy12] are some of the main features that have been studied during the last years in the DG framework. More recently, Space-Time Discontinuous Galerkin methods (STDG) have emerged. These formulations exploit a temporal discretization similar to the spatial one instead of exploiting classical advancing-in-time schemes like LF or RK.

The following section presents the non-dissipative DGTD method originally developped by Fezoui et al. in [FLLP05]. Let Ωh be a discretization of Ω, relying on a quasi-uniform triangulation_ThverifyingTh=SNi=1Ti. The internal faces of the discretization are denoted

aik = TiTTk, and nikis defined as the unit normal vector to the faceaik, oriented fromTi towardTk. For each cellTi,Viis the set of indices{k | TiTTk 6= ∅}. Then, the quasi-uniform assumption implies that :

∀Ti∈ Th,∀k ∈ Vi,∃δ, hk ≤ δhhi.

The semi-discrete fields are denoted(Hh,Eh,Jh), and on each cellTithe restrictions(Hi,Ei,Ji) = Hh Ti,Eh Ti,Jh Ti

_{are defined.}

For each cell, a set of scalar basis functions(φij)_16j6diis defined, wherediis the number of degrees of freedom (d.o.f.) per dimension (therefore, a 3D problem implies3 di d.o.f. per cell).

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2.1 Weak formulation

It is now possible to write the weak formulation of the problem (8) in a cellTi. By taking the dot-product of each term with a vectorial test functionψand then integrating over the cell, one obtains the following variational problem :

Find(E,H)_{∈ H}0(rot,Ω)× H (rot,Ω) such that∀ ψ ∈ H (rot,Ω) ,

       Z Ti µ0 ∂H ∂t · ψ + Z Ti∇ × E_{· ψ =} 0, Z Ti ε0 ∂E ∂t · ψ − Z Ti∇ × H_{· ψ = −} Z Ti J_{· ψ,}

which can be rewritten as follows using classical vectorial calculus and Green formulae :

       Z Ti µ0 ∂H ∂t · ψ + Z Ti E_{· ∇ × ψ =} Z ∂Ti (ψ_×E)_·ni, Z Ti ε0 ∂E ∂t · ψ − Z Ti H_{· ∇ × ψ = −} Z Ti J_{· ψ −} Z ∂Ti (ψ_×H)_·ni.

One immediatly notices that the previous equality only holds if the boundary terms exist. Considering the properties of the mixed product, the latter becomes :

(ψ_×E)_·ni= (E×ni)· ψ,

which implies that taking E inH0(rot,Ω)requires the normal and tangential trace ofψon

∂Tito exist. This implies that one should takeψinH1(Ω)instead ofH(rot,Ω).

Remark :

A set ofdi scalar basis functionsφij have previously been defined on the cellTi. Nevertheless,

the test functions are to be naturally chosen as vectorial elements. Therefore, one should define three vectorial basis functions (for the three space dimensions) for each scalar one. We now denote :

φ1ij=   φij 0 0   , φ2 ij=   0 φij 0   andφ3ij=   0 0 φij   .

2.2 Space discretization

2.2.1 Volumic integrals

We now seek the approximations Ehand Hhof E and H in the following approximation space : Vh= n v_{∈ L}2(Ω)3, vTi ∈ (Pp(Ti))3∀ Ti∈ Th o , (9)

whereP_p_(T_i₎is the space of polynomials of maximum degreeponTi. The contribution of each cell is therefore defined as E_i=Eh Ti, where Eiis locally expanded as :

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Ei= 3 X d=1 di X j E_ijdφd_ij

We now choose the test functionsψ equal to the3 di vectorsφdij. The spatial discretization of

Z

Ti

ε0

∂E

∂t · ψthen leads to3 diterms of the following form : Z Ti ε0Tφdij· φ d il ∂Ed ij ∂t . (10)

The previous term can be cast under a matrix formMε0

i Ei, where the mass matrix is block diagonal of size3 di× 3 di: Mε0_i =    e Mε0_i 0 0 0 Meε0 i 0 0 0 Meε0_i    , with e Mε0_i jl = Z Ti ε0Tφdij· φ d il

The vector Eiintroduced previously has3 dicomponents, and is defined as follows :

Ei=    E_ij1 16j6di E_ij2_16j6d i E_ij3_16j6di   

From now on, the mass matrices with an exponent should be understood as :

e Mx_i jl= Z Ti xTφd_ij_{· φ}d_il,

and their associated block matrices denotedMx_i. 2.2.2 Surface integrals

Given that E and H are discontinuous at cell boundaries, the surface integrals need a special treatment. We choose a centered approximation, which permits to rewrite

Z ∂Ti (ψ×E)·n as : X k∈Vi Z aik φd_ij _×Ei+Ek 2 ·nik, with a similar choice for the surface integral involving H. 2.2.3 Semi-discrete equations

Taking into account the previous definitions, one obtains the following6 discalar equations :

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22 Lanteri, Scheid & Viquerat            Mµ0_i ∂Hi ∂t j + Z Ti Ei· ∇ × φdij = X k∈Vi Z aik φd_ij_×Ei+Ek 2 ·nik, Mε0_i ∂Ei ∂t j − Z Ti Hi· ∇ × φdij = − Z Ti J_i_{· φ}d ij− X k∈Vi Z aik φd_ij_×Hi+Hk 2 ·nik. (11)

Performing an integration by parts on the surface integrals leads to :

                     Mµ0 i ∂Hi ∂t j = −1₂ Z Ti Ei· ∇ × φdij+∇ ×Ei· φdij +1 2 X k∈Vi Z aik φd_ij_{· (}Ek×nik) , Mε0 i ∂Ei ∂t j = 1 2 Z Ti Hi· ∇ × φdij+∇ ×Hi· φdij − Z Ti J_i_{· φ}d ij− 1 2 X k∈Vi Z aik φd_ij_{· (}Hk×nik) . (12)

Reasoning similarly as for (10) permits to rewrite (12) as :

           Mµ0 i ∂Hi ∂t = −KiEi+ X k∈Vi S_ikEk, Mε0 i ∂Ei ∂t = KiHi− X k∈Vi S_ikHk− MiJi,

where the3 di× 3 dilocal stiffness matrix and the3 di× 3 dksurface matrix are respectively defined by : e K_i jl= 1 2 Z Ti φd_ij_{· ∇ × φ}d_il+ φd il· ∇ × φ d ij and eSik jl= 1 2 Z Ti φd_ij_{· φ}d_kl_×nik. As mentioned previously, the basis functions are chosen in the approximation spaceVh. A common choice consists in using Lagrange polynomials, though other choices are possible [CFL10]. A set ofpi+ 1interpolation nodes(xj)06j6pi is defined in the cell, and the basis functions are then the Lagrange functionLp_k(x)equal to 1 on the xk node, and to 0 on all the otherxj, j6= k.

2.3 Time discretization

For the time derivatives, a second-order leap frog scheme (LF2) is used, where the E_iare evaluated at the time stationtn_{= n∆t}_{, whereas the H}

i,Jiare evaluated at the time station

tn+12 = n +1 2

∆t. This leads to seek the values of En+1 i and H

n+3 2

i when knowing those of En i, H n+1 2 i and J n+1 2

i with the following discretization :

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           Mµ0_i ∆t Hn+32 i −H n+1 2 i = −KiEn+1i + X k∈Vi S_ikEn+1_k , Mε0 i ∆t E n+1 i −E n i = KiH n+1 2 i − X k∈Vi S_ikHn+12 k −J n+1 2 i . (13)

Remark : Different time discretization can be used, like fourth-order leap frog (LF4) scheme or second

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3 DGTD formulation in dispersive media

The case of Maxwell’s equations in vacuum has been presented in the previous section. After a brief review of existing works, the extension of the DGTD formulation to Drude-like media is presented. Then, the same treatment is done for the generalized dispersive model.

3.1 Previous works

Before they were exploited with DGTD methods, dispersion models have been extensively used in the context of FDTD, and a very large amount of references are therefore available on this topic. We only settle here for giving a very few of them, such as [OO06] that presents an ADE-FDTD (Auxiliary Differential Equation) algorithm with the Drude model, or [LSK93] that uses a RC-FDTD (Recursive Convolution) method for Debye, Drude and Lorentz models. Although the exploitation of dispersion models in the DGTD framework is not as rich as the FDTD one, an important amount of studies have already been conducted. In [LS12], the authors study a DGTD-CF (Centered Fluxes) method for the Maxwell equations coupled with a Debye dispersion model. Stability and convergence are proved, and a bound on the error is given. In [JCZ07], the authors study the 2D Maxwell equations in a Drude-like medium with a DGTD-CF of the fourth order in space, and with a RK4 scheme in time.

A significant number of contributions on the numerical analysis of the schemes for dis-persive media models have been made by J. Li, such as [LCE08] and [Li09] to name a few. Numerous applications of the DG method in the area photonics have been issued : [BKN11] presents a DGTD and DGFD (Discontinuous Galerkin Frequency Domain) method as well as realistic cases, and discuss topics like sources treatment and boundary conditions, like ab-sorbing ones (ABC). In [SKNB09], the authors focus on the field enhancements observed in the vicinity of metallic V-shaped nanostructures described by Drude model. A nice overview of the DGTD method coupled with Drude and Drude-Lorentz models can be found in [Die12], where the author exploit a GPU implementation to focuses on the analysis of various nanos-tructures features.

3.2 DGTD method in Drude-like media

3.2.1 Maxwell-Drude equations

We now consider the case of a frequency-dependent medium, under the hypothesis of a Drude-Sommerfeld model :

εr(ω) = ε∞−

ω2 d

ω2_{+ iωγ}.

Considering a constant permeability and a homogeneous and isotropic medium, one writes the general Maxwell equations as follows :

     ∇ ×E = ₋∂B ∂t, ∇ ×H= ∂D ∂t, (14)

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(

D = ε0ε∞E+P, B = µ0H.

(15) Combining (14) and (15) leads to :

     ∇ ×E = _−µ0 ∂H ∂t, ∇ ×H = ε0ε∞ ∂E ∂t + ∂P ∂t.

In the frequential domain, and under the aforementioned hypothesis, the polarization P is linked to the electric field throughPˆ =₋ ε0ω2d

ω2_+iγ_dωE, whereˆ ˆ· denotes the Fourier transform of the associated field in the temporal domain. An inverse Fourier transform gives :

∂2P ∂t2 + γd ∂P ∂t = ε0ω 2 dE. (16)

By defining the dipolar current vector J_p=∂P

∂t, (14)-(16) can be rewritten as follows :              µ0 ∂H ∂t = −∇ ×E, ε0ε∞ ∂E ∂t = ∇ ×H−Jp, ∂J_p ∂t + γdJp = ε0ω 2 dE. (17) 3.2.2 Normalization

The previous set of equations is now normalized. A normalized variable is denotedXe when the original variable isX. First, we define vacuum impedance and light velocity :

Z0= r µ0 ε0 andc0= 1 √_ε 0µ0 .

Then, the following substitutions are applied :

e H= Z0H, eE=E, eJp= Z0Jp, et= c0t,eγd = γd c0 andω_e2_d= ω 2 d c2 0 .

One can now rewrite (17) in the form :

                 µ0c0 Z0 ∂ eH ∂et = −∇ × eE, ε0c0Z0ε∞ ∂ eE ∂et = ∇ × eH− eJp, ∂eJ_p ∂et + eγdeJp = Z0ε0c20 c0 e ω2_dEe.

Using the equalities µ0c0

Z0

= 1andε0c0Z0= 1, one gets :

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                 ∂ eH ∂et = −∇ × eE, ∂ eE ∂et = 1 ε∞ ∇ × eH_{− e}J_p, ∂eJ_p ∂et = eω 2 dEe− eγdeJp. (18)

It could be noticed that the first two equations are expressed in V.m−2, whereas the third one is inV.m−3_.

Remark : To simplify the writings, theXe notation for the normalized variables will be omitted from

now on.

3.2.3 DGTD formulation for the Maxwell-Drude equations

The extension of the DGTD method to the Maxwell-Drude equations is straightforward from the classical Maxwell equations discretization (see section2). The last equation of (18) leads to a simple vectorial equation :

1 ∆t Jn+ 3 2 i − J n+1 2 i =_−γdJ n+1 2 i + ω 2 dEn+1i . The whole system can then be written as :

                   M_i ∆t Hn+32 i −H n+1 2 i = −KiEn+1i + X k∈νi S_ikEn+1_k , Mε∞ i ∆t E n+1 i −E n i = KiH n+1 2 i − X k∈νi S_ikHn+12 k − MiJ n+1 2 i , 1 ∆t Jn+32 i − J n+1 2 i = ω2 dE n+1 i − γd 2 Jn+32 i +J n+1 2 i . (19)

3.3 DGTD method for the generalized dispersive model

A similar development as the one of the previous section can be made for the general per-mittivity function (7).

3.3.1 Continuous equations

As in the Drude case, polarizations and currents are introduced to account for the dispersive behavior in the temporal domain. The (normalized) continuous equations are therefore :

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28 Lanteri, Scheid & Viquerat                                              ∂H ∂t =−∇ ×E, ∂E ∂t = 1 ε∞ ∇ × H₋ J0+ X l∈L1 J_l+X l∈L2 J_l !! , J₀ = σE, Jl = alE− blPl ∀l ∈ L1, ∂Pl ∂t =Jl ∀l ∈ L1, ∂J_l ∂t = dl ∂E ∂t + clE− flJl− elPl ∀l ∈ L2, ∂Pl ∂t =Jl ∀l ∈ L2. (20) 3.3.2 DGTD formulation

In this formulation, the currents are evaluated at the even time stations, whereas the po-larizations are evaluated at the odd ones. The resulting scheme in time is, as well as in the Drude case, of order two (see section5.2). As one might notice, some fields have been substituted to avoid unnecessary calculations :

                                                                   M_i ∆t Hn+32 i − H n+1 2 i = −KiEn+1i + X k∈νi S_ikEn+1_k , M_i ∆t E n+1 i −E n i = 1 ε∞ K_iHn+12 i − X k∈νi S_ikHn+12 k − σM₂i En+1_i + Eni + Mi X l∈L1 blP n+1 2 l,i −M₂i X l∈L2 Jn+1_l,i +Jn_l,i ! , 1 ∆t Pn+32 l,i − P n+1 2 l,i = alEn+1i − bl 2 Pn+32 l,i + P n+1 2 l,i ∀l ∈ L1, 1 ∆t Pn+32 l,i − P n+1 2 l,i = Jn+1_l,i _{∀l ∈ L}2, 1 ∆t Jn+1_l,i ₋Jn_l,i = dl ∆t E n+1 i −Eni +cl 2 E n+1 i + Eni , −f₂lJn+1_l,i + Jn l,i − elP n+1 2 l,i ∀l ∈ L2. (21) where σ = σ +X l∈L1

al. The resulting updating scheme is presented in figure4.

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En Hn+12 Jn_L2 _Pn+12 L2 Pn+12 L1 En+1 Hn+32 Jn+1_L2 Pn+32 L2 Pn+32 L1

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4 Theoretical study of the Maxwell-Drude equations

A stability and convergence study of the Maxwell-Drude equations is now presented. This part is highly inspired from [LS12].

4.1 Stability study of the Maxwell-Drude equations

4.1.1 Continuous equations

First, the energy associated to the differential system (18) is defined at a given timet:

ξ(t) = 1 2 kH(t)_k2_L2_(Ω)+ ε∞kE(t)k2L2_(Ω)+ 1 ω2 d Jp(t) 2 L2_(Ω) . (22)

Assuming that(E,H)have a sufficient regularity (i.e. (E,H) ∈ H0(rot,Ω)× H (rot,Ω)in space, andC1in time, for example), one would like to prove that this energy decreases with time. TheL2scalar product onΩof each equation leads to :

∂ξ ∂t =− Z Ω (_{∇ ×}E)_·H+ Z Ω (_{∇ ×}H)_·E₋ γd ω2 d Z Ω Jp·Jp. The Green formula applied to the first term implies :

Z Ω (_{∇ ×}E)_·H= Z Ω (_{∇ ×}H)_·E₋ Z ∂Ω (H_×E)_·n.

Using the property of the mixed product, and given the metallic boundary conditions, the boundary integral is equal to zero. Therefore :

∂ξ ∂t =− γd ω2_d J_p(t) 2_L2_(Ω)≤ 0 (23) 4.1.2 Semi-discrete case

We now seek the approximated fields in the Vh space that has been previously defined in equation (9). Moreover,_∀Wh∈Vh, the following notations are used :

• The restriction W_i=Wh Ti,

• The mean of W_hacross an interface between two cellsTiandTk :

{Wh}ik=

Wi aik+Wk aik

2 ,

• The tangential jump of Whthrough an interface :

JWhKik= Wk aik−Wi aik

×nik.

We focus on the differential system (18). Taking theL2_{scalar product on}_T

iof each equation with test functionsϕ, ψ, φsufficiently regular leads to :

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32 Lanteri, Scheid & Viquerat                  Z Ti ∂H ∂t · ϕ = − Z Ti E_{· (∇ × ϕ) +} Z ∂Ti ϕ_{· (}E_×n) , ε∞ Z Ti ∂E ∂t · ψ = Z Ti H_{· (∇ × ψ) −} Z ∂Ti (ψ_×H)_·n₋ Z Ti J_p_{· ψ,} 1 ω2 d Z Ti ∂J_p ∂t · φ = Z Ti E_{· φ −} γd ω2 d Z Ti J_p_{· φ.}

One defines the semi-discrete fields(Hh,Eh,Jh)as solutions of the following weak formula-tion :_{∀ (ϕ}_h, ψh, φh)∈V3h,∀t ∈ [0, T ], ∀i ∈ [0, NT],                    Z Ti ∂Hh ∂t · ϕh = − Z Ti Eh· (∇ × ϕh) + X k∈Vi Z aik ϕh· ({Eh}_ik×n) , ε∞ Z Ti ∂Eh ∂t · ψh = Z Ti Hh· (∇ × ψh)− X k∈Vi Z aik (ψh× {Hh}ik)·n− Z Ti J_h_{· ψ}h, 1 ω2 d Z Ti ∂Jh ∂t · φh = Z Ti Eh· φh− γd ω2 d Z Ti J_h_{· φ}h. (24)

and the associated semi-discrete energyξhis therefore defined, according to the definition ofξ, as follows : ξh= 1 2 kHhk 2 L2_(Ω)+ ε∞kEhk 2 L2_(Ω)+ 1 ω2 d kJ_h_k2_L2_(Ω) .

Taking(ϕh, ψh, φh) = (Hh,Eh,Jh), and then summing the three equations of (24) together and over all cells, one gets :

∂ξh ∂t = Z Ω (Hh· (∇ ×Eh)−Eh· (∇ ×Hh))− γd ω2 d Z Ω J_h_·J_h + Z Fint{ Hh} JEhK− Z Fint{ Eh} JHhK + Z ∂Ω Eh· (Hh×n) . A final integration by parts6_{leads to :}

∂ξh ∂t =− γd ω2 d kJ_h_k2_L2_(Ω). (25) Hence, ξh(t) ≤ ξh(0), and the stability of the semi-discrete scheme is ensured. For the rest of this study, we need to write the semi-discrete formulation over the whole domain. Therefore, let U= (X, Y, Z)and U′ _{= (X}′_{, Y}′_{, Z}′₎_{, and let the following bilinear forms :}

m U,U′ = Z Ω X_{· X}′+ ε∞ Z Ω Y _{· Y}′+ 1 ω_d2 Z Ω Z_{· Z}′ a U,U′ = Z Ω (X_{· (∇ × Y}′)_{− Y · (∇ × X}′))₋ Z Ω Z_{· Y}′+ Z Ω Y _{· Z}′₋ γd ω2 d Z Ω Z_{· Z}′ b U,U′ = Z Fint {X} JY′K₋ Z Fint {Y } JX′K + Z ∂Ω Y′_{· (X ×}n) .

6_{Realized on each cell}_T

iand then summed over the whole domain.

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Let Uh∈Vhbe the solution of the semi-discrete problem. Given thee latter definitions, the semi-discrete formulation can be rewritten on the entire domain as :

m ∂Uh ∂t ,U ′ h = a Uh,U′h + b Uh,U′h ,_∀Uh∈V6h. (26) Moreover, the solution U of the continuous equations verifies :

m ∂U ∂t,U ′ h = a U,U′_h+ b U,U_h′,_∀Uh∈V6h. (27)

4.1.3 Fully discrete scheme

A second-order leap-frog (LF2) time discretization is chosen. The electric field on one hand, and the magnetic field and polarization current on the other hand are evaluated respectively at the even and odd time stations. The fully discrete scheme can then be written as follows :                                  Z Ti Hn+32 i −H n+1 2 i ∆t · ϕh = − Z Ti En+1_i _{· (∇ × ϕ}h) + X k∈Vi Z aik ϕh· En+1_h ik×nik , ε∞ Z Ti En+1_i ₋Eni ∆t · ψh = Z Ti Hn+ 1 2 i · (∇ × ψh)− X k∈Vi Z aik ψh× n Hn+ 1 2 i o ik ·nik − Z Ti Jn+12 i · ψh, 1 ω2 d Z Ti Jn+ 3 2 i −J n+1 2 i ∆t · φh = Z Ti En+1_i _{· φ}h− γd ω2 d Z Ti Jn+ 3 2 i +J n+1 2 i 2 · φh, (28)

and we define the associated energy on the cellTiby :

ξ_in= 1 2 Z Ti Hn+12 i ·H n−1 2 i + ε∞ Z Ti En_i _·En_i + 1 ω2 d Z Ti Jn+12 i ·J n−1 2 i (29) The total energy at a given timetn= n∆tis calculated as :

ξn=

NT

X

i=0

ξ_in

Since they will be useful later, we remind to the reader the following classical inequalities [BS08].

Lemma 1. letTibe a cell of the meshTh. Then,∃C > 0,such that∀ u ∈ Pp(Ti):

k∇ × ukTi≤

C

h kukTi, (30)

and

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where_k·k_Tiand_k·k_aik represent theL2norms defined respectively on the cellTiand on the

face of the cellaik.

Proposition 1 (Stability). The formulation (28) is stable under the following condition :

∆t < min h C, 2 ωd+ γd , _C4ε∞ h − ωd ! (32)

Proof. One would like to know under which conditions the energyξn _{can be written as a} definite positive form of the variables Hn−1

2, En and Jn− 1

2. To do so, we seek a lower bound ofξn_{by using the equations of the system (}₂₈_{) to replace the occurences of H}n+1

2 and Hn− 1 2 with the help of well-chosen test functions. More accurately, the following substitutions are used :

• The first equation of (28) is used at timetnwithϕh=H n−1

2 i ;

• The second one attnwithψh=Eni;

• The third one attnwithφh=J n−1 2 i . These lead to : ξn i = 1 2 Hn−12 i 2 Ti+ ε∞ En_i 2 Ti+ α βω2_d Jn−12 i 2 Ti− ∆t Z Ti ∇ ×Hn−12 i ·En_i + ∆t β Z Ti Eni ·J n−1 2 i + ∆t X k∈Vi Z aik Hn− 1 2 i · ({E n h}ik×nik) ! , whereα= 1−∆tγd2 andβ = 1 + ∆tγd

2 . Splitting the curl term in two parts, and integrating by parts one out of the two, one gets (the power indices are omitted up to the end of this proof) : ξn i = 1 2 kHik 2 Ti+ ε∞kEik2Ti+ α βω_d2kJik 2 Ti − ∆t₂ Z Ti (Ei· (∇ ×Hi) +Hi· (∇ ×Ei)) + ∆t β Z Ti Ei·Ji + ∆tX k∈Vi Z aik Hi· ({Eh}_ik×nik)− ∆t 2 X k∈Vi Z aik (Hi×Ei)·nik ! .

The surface integrals then rearrange as follows, given the metallic boundary conditions at the boundaries of the domain :

∆tX k∈Vi Z aik Hi· ({Eh}ik×nik)− ∆t 2 X k∈Vi Z aik (Hi×Ei)·nik = ∆t 2 X k∈FintTVi Z aik Hi· (Ek×nik)− ∆t 2 X k∈∂ΩTVi Z aik Hi· (Ei×nik) . Inria

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The different integrals are then bounded as follows using (30) and (31) : Z Ti (Ei· (∇ ×Hi) +Hi· (∇ ×Ei)) ≤ C hkEikTikHikTi, Z aik Hi· (Ei×nik) ≤ C hkEikTikHikTi, Z Ti Ei·Ji ≤ kEikTikJikTi.

Using the previous relations as well as the classical inequalityab_≤ 1₂ a2+ b2_{, one gets :}

ξin ≥ 1 2 kHik2Ti+ ε∞kEik2Ti+ α βω2 d kJik 2 Ti − ∆t₈ C_hkEik2_Ti+kHik2_Ti −∆t₄ ω_βd kEik2_Ti+ 1 ω2 d kJ_i_k2_Ti (33) − ∆t₈ C_h X aik∈Fint kHik2_Ti+kEkk2_Tk + X aik∈∂Ω kHik2_Ti+kEik2_Ti ! .

In order to make the CFL conditions more readable, it is assumed that :

∆t_≤ 2 γd , (34) which implies : 1 > 1 β > 1 2.

Therefore, adjustingC, (33) can be rewritten as :

ξ_in _≥ 1 2 1₋C∆t 4h kHik2Ti+ 1 2 ε∞− C∆t 4h − ∆tωd 4 kEik2Ti − C∆t_8h X k∈Vi kEkk2Tk+ 1 4ω2 d 1₋γd∆t 2 − ωd∆t 2 kJ_i_k2_Ti.

Then, summing over all the cells and adjustingC, one obtains an inequality involvingξn _:

ξn_≥ 1 2 1−C∆t_h kH_k2_Ω+1 2 ε∞−∆t 4 C h − ωd kE_k2_Ω+ 1 4ω2 d 1−∆t₂ (γd+ ωd) kJ_k2_Ω.

Each one of the three induced conditions is now considered separately :

∆t < h C, ∆t < 4ε∞ C h − ωd , ∆t < 2 ωd+ γd .

One should eventually notice that the condition (34) is contained in the last condition above-written, which leads to the desired result.

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Proposition 2 (Bound of the discrete energy). Under CFL condition, the discrete energy

ξncan be bounded in the following way :

ξn_≤ ξ 0 1−θ 1+θ n ∀n ∈ N∗ withθ_{≥ 0}.

Proof. In order to bound the discrete energy, one would first be interested in knowing the

sign ofξn+1_{− ξ}n_,_{∀ n ≥ 0}_{. To do so, the equations of (}₂₈_{) are used at different times and for} different test functions :

• The first one at timetnand timetn+1withϕh=H n+1

2 i ;

• The second one at timetnwithψh=E[ n+1

2]

i ;

• The last one at timetnand timetn+1withφh=J n+1

2 i . Combining these different expressions yields :

ξn+1_i _{− ξ}_in = ∆t Z Ti Hn+ 1 2 i · ∇ ×E[n+ 1 2] i −E[n+ 1 2] i · ∇ ×Hn+ 1 2 i + ∆tX k∈Vi Z aik Hn+ 1 2 i · E[n+ 1 2] i ×nik −E[n+ 1 2] i · Hn+ 1 2 i ×nik + ∆t Z Ti Jn+ 1 2 i · ₁ 2βE n+1 i + 1 2αE n i −E [n+1 2] i −α 2 − β2 2αβω2 d Z Ti Jn+ 1 2 i ·J n+1 2 i . Integrating by parts the curl terms twice yields :

ξ_in+1_{− ξ}n i = − ∆t 2 X k∈Vi Z aik E[n+ 1 2] i ×H n+1 2 k +E [n+1 2] k ×H n+1 2 i ·nik + ∆t 2_γ d 4 Z Ti Jn+ 1 2 i · − En+1_i 1 + ∆tγd₂ + En_i 1₋∆tγd₂ ! | {z } η + 1 ω2 d ∆tγd ∆t2_γ2 d 4 − 1 | {z } θ0 Jn+ 1 2 i 2 Ti. If∆t≤ 2 γd, thenθ0<0and|θ| < |θ1| = ∆t2 γ22_d 4 −1

. One can obtain the following bound forη: |η| ≤ ∆t₄ |θ1| β Jn+12 i TikE n ikTi+ α Jn+12 i Ti En+1_i _Ti ≤ ∆tω₂ d |θ1| β Jn+12 i Ti ωd max_kEn_i_k_Ti, En+1_i _Ti | {z } ρi ≤ ∆tω₂ d |θ1|    Jn+12 i 2 Ti ω2 d + ρ2 i    . Inria

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whereα= 1 +∆tγd

2 andα= 1− ∆tγd

2 . As specified earlier, if∆t≤

2

γd, then one has :

β_{≤ 2}andα_{≤ 1.}

Then, by summing the contributions over all the mesh cells, and by considering metallic boundaries, it is possible to write :

ξn+1_{− ξ}n _≤ NT X i=0      ∆tωd|θ1| 2    Jn+ 1 2 i 2 Ti ω_d2 + ρ 2 i    +_ωθ02 d Jn+12 i 2 Ti | {z } ≤0      ≤ NT X i=0 |θ1|∆tωd 2    Jn+12 i 2 Ti ω2 d + ρ2i    .

LetC be a generic constant. Under CFL condition, the first term involving NT X i=1 Jn+ 1 2 i 2 Ti is

bounded byCξn+1, whereas the NT

X

i=1

ρ2_i one is bounded byC ξn_{+ ξ}n+1_{. This yields :}

ξn+1_{≤ |θ}1| ∆tωdξn+1+ 1 +|θ1| ∆tωd 2 ξn.

The latter leads to :

ξn+1_≤ 1 + |θ1|∆tωd 2 1_{− |θ}1| ∆tωd ξn_≤ 1 +|θ1| ∆tωd 1_{− |θ}1| ∆tωd ξn= 1 + θ 1_{− θ}ξ n_,

withθ_{≥ 0}. It is also required thatθ_{≤ 1}, which can be shown to be equivalent to :

∆t≤γ 2 d 2 −2ωd+ q 4ω2 d+ γd2 .

The latter condition can be proved to be less restrictive than the∆t _≤ 2

γd. Therefore, we have obtained the expected result.

4.2 Convergence of the fully discrete Maxwell-Drude DG formulation

In this section, the convergence of the Maxwell-Drude DG scheme is proven, after a prelim-inary convergence result about the semi-discrete formulation. We start with the following lemma.

Lemma 2. Let Ti be a cell of the mesh Th, and πh(·)a linear continuous projector from

Hs+1(Ti)ontoPp(Ti)withs≥ 0andk≥ 1. Then, foru∈ Hs+1(Ti)andm= 0, 1:

|u − πh(u)|m,Ti≤ Ch

min(s,p)+1−m

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ku − πh(u)k0,∂Ti ≤ Ch

min(s,p)+1 2

Ti kuks+1,Ti. (36)

4.2.1 Convergence of the semi-discrete formulation

Theorem 1 (Convergence of the semi-discrete formulation). Let H,E,J_pbe the solution of (17) and (Hh,Eh,Jh) ∈ C1 [0, T ] ,V3h

the semi-discrete solution of (24). If H,E,J_p _∈

C0_{[0, T ] , H}s+1_(Ω)9

fors_{≥ 0}, then there existsC_{≥ 0}independent ofhsuch that :

max t∈[0,T ]γ(t) 1 2 ≤ Chmin(s,p) _H,_E,_J p C0₍_{[0,T ],H}s+1_(Ω)9₎, with : γ(t) =_kπh(H)−Hhk2L2_(Ω)+kπh(E)−Ehk2L2_(Ω)+ πh Jp −J_h 2 L2_(Ω)∀t ∈ [0, T ] .

Proof. The orthogonalL2projection on the V3

hspace is defined asπh(U) = (πh(H) , πh(E) ,

πh Jp

_{, and the consistency error as}

ε(t) = 1

2m(πh(U)−Uh, πh(U)−Uh). Then, it is easy to boundξin the following way :

ε(t)_≥1 2min 1, ε∞, 1 ω2 d kπh(H−Hh)k2L2_(Ω)+kπh(E−Eh)k2L2_(Ω)+ πh Jp−Jh 2 L2_(Ω)

In addition, given thatπh(U)−Uh∈ Vh, one could easily prove that :

m πh ∂U ∂t −∂_∂tU, πh(U)−Uh = 0, (37) a πh ∂U ∂t −∂_∂tU, πh(U)−Uh = 0. (38)

Sincem,aandbare bilinear forms, and given the equalities (26-27-37-38), it is possible to write (the proof is exactly the same as in [LS12], pp 12 – 13) :

m ∂πh(U) ∂t − ∂Uh ∂t , πh(U)−Uh = a(πh(U)−Uh, πh(U)−Uh) + b(πh(U)−Uh, πh(U)−Uh) + b(U_{− π}h(U) , πh(U)−Uh) .

Given what has already been proven in the stability study of the semi-discrete formulation, it is straightforward to prove that we have :

a(πh(U)−Uh, πh(U)−Uh) + b (πh(U)−Uh, πh(U)−Uh) =− γd ω_d2 πh Jp −J_h 2 Ω. Then, the termb(U_{− π}h(U) , πh(U)−Uh)remains to be bounded. Let us write :

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b(U_{− π}h(U) , πh(U)−Uh) = Z Fint {H_{− π}h(H)} Jπh(E)−EhK | {z } ζ − Z Fint {E_{− π}h(E)} Jπh(H)−HhK | {z } µ + Z ∂Ω (πh(E)−Eh)· ((H− πh(H))×n) | {z } κ .

First, one considers theζterm. Applying the Cauchy-Schwarz inequality leads to :

|ζ| ≤ X aik∈Fint Z aik{ H_{− π}h(H)}ikJπh(E)−EhKik ≤ X aik∈Fint Z aik{ H_{− π}h(H)}ikJπh(E)−EhKik ≤ X aik∈Fint Z aik|{ H_{− π}h(H)}_ik|2 1 2Z aik|Jπ h(E)−EhKik|2 1 2 .

The first sum can be bounded as follows :

Z aik|{ H_{− π}h(H)}_ik|2 ≤ C kHi− πh(Hi)k2_L2_(aik)+kHk− πh(Hk)k2_L2_(aik) ≤ CkHi− πh(Hi)k2L2_(∂Ti)+kHk− πh(Hk)k2L2_(∂Tk) ≤ Chmin(s,p)+ 1 2 Ti kHks+1,Ti 2 +hmin(s,p)+ 1 2 Tk kHks+1,Tk 2 ≤ Cδ2 min(s,p)+1_h2 min(s,p)+1 Ti kH_k2_s+1,Ti+kH_k2_s+1,T k ,

the last inequality resulting from the quasi-uniform assumption made on the triangulation of the domain. Bounding the second integral can be done almost the same way, and requires the inverse inequalities presented earlier :

Z aik|Jπ h(E)−EhKik|2≤ Cδ−1h−1Ti kπh(E)−Ehk20,Ti+kπh(E)−Ehk20,Tk .

Then, by adjusting the constantC, one obtains :

|ζ| ≤ Cδmin(s,p) X (i,k)|aik∈Fint hmin(s,p)_Ti _kH_k2_s+1,Ti+_kH_k2_s+1,Tk 1 2 kπh(E)−Ehk20,Ti+kπh(E)−Ehk20,Tk 1 2 X 2 2 12 ₂ ₂ 12

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whereh= max

Ti hTi. Eventually, one must consider the sum with respect to all the internal faces, which can be rewritten as follows by adjusting once again the generic constantC:

|ζ| ≤ Cδmin(s,p)hmin(s,p)X

Ti

kH_k_s+1,Ti_kπh(E)−Ehk0,Ti. Boundingµcan be done the exact same way :

|µ| ≤ Cδmin(s,p)hmin(s,p)X

Ti

kE_k_s+1,Ti_kπh(H)−Hhk0,Ti.

Moreover, one could easily prove thatκ= 0, given the metallic boundary condition on∂Ω. Therefore :

|ζ| + |µ| ≤ X

Ti

Cδmin(s,p)hmin(s,p)_Ti _kH_k_s+1,Ti_kπh(E)−Ehk0,Ti+kEks+1,Tikπh(H)−Hhk0,Ti

≤ C max Ti δ min(s,p) hmin(s,p)_Ti   X Ti kH_k2_s+1,Ti !1 2 X Ti kπh(E)−Ehk20,Ti !1 2 + X Ti kE_k2_s+1,Ti !1 2 _X Ti kπh(H)−Hhk20,Ti !1 2  ≤ Cδmin(s,p)hmin(s,p)_kH_k_s+1,Ω_kπh(E)−Ehk0,Ω+kEks+1,Ωkπh(H)−Hhk0,Ω ≤ Cδmin(s,p)hmin(s,p)_k(H,E)_k_s+1,Ω_kπh(E)−Ehk20,Ω+kπh(H)−Hhk20,Ω 1 2 . Sinceε(t) = 1₂ Z t 0 m ∂(πh(U)−Uh) ∂s , πh(U)−Uh

ds, and assuming thatε(0) = 0, one can write : ε(t) = 1 2 Z t 0 − γd ω2 d kJ_h_k2_Ωds+1 2 Z t 0 b(U_{− π}h(U) , πh(U)−Uh)ds ≤ 1₂ Z t 0 b(U_{− π}h(U) , πh(U)−Uh)ds. In regard of the previous results, one obtains :

ε(t) _{≤ Ch}min(s,p) Z t 0 k( H,E)_k_s+1,Ω_kπh(E)−Ehk20,Ω+kπh(H)−Hhk20,Ω 1 2 ds ≤ Chmin(s,p) Z t 0 H,E,J_p _s+1,Ωγ(s)12ds, therefore : γ(t)_{≤ Ch}min(s,p)υ H,E,Jp _C0₍_{[0,T ],H}s+1_(Ω)3₎, whereυ= max[0,T ]γ(t) 1

2. Taking the maximum value of the left hand side over[0, T ]permits to conclude.

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4.2.2 Convergence of the fully discrete formulation

We consider the fully discrete formulation presented previously in equation (28). We choose a constant time step∆tverifying the CFL condition (32) and such thatN∆t = T, whereNis the total number of timesteps. Two classical results about Taylor-Lagrange expansions will be required : Lemma 3. Let beU _{∈ C}3_([t n, tn+1]), then∃ (cn, cn+1)∈]tn, tn+1 2[×]tn+ 1 2, tn+1[and(dn, dn+1)∈ ]tn, tn+1 2[×]tn+ 1 2, tn+1[such that : Uh(tn+1)− Uh(tn) = ∆t ∂Uh ∂t tn+1 2 +∆t 3 28 ∂3Uh ∂t3 (cn+1) + ∂3Uh ∂t3 (cn) , and 1 2(Uh(tn+1) + Uh(tn)) = Uh t_n+1 2 +∆t 2 16 ∂2Uh ∂t2 (dn+1) + ∂2Uh ∂t2 (dn) .

Theorem 2 (Convergence of the fully discrete formulation). Let be : H,E,J_p_{∈ C}3[0, T ] , L2(Ω)9 \_C0[0, T ] , Hs+1(Ω)9.

Under a CFL condition as in (32), the following error estimate holds :

max n∈[0,N ] Htn+1 2 −Hn+ 1 2 h 2 L2_(Ω)3+kE(tn)−E n hk 2 L2_(Ω)3 + J_ptn+1 2 −Jn+ 1 2 h 2 L2_(Ω)3 1 2 ≤ C∆t2+ hmin(s,k) H,E,J_p _C3₍_{[0,T ],L}2_(Ω)9_{) +} H,E,J_p _C0₍_{[0,T ],H}s+1_(Ω)9₎

Proof. We define the consistency error as follows :

εn+1_h = Eh(tn+1)− eE n+1 h 2 L2_(Ω)+ Hh t_n+3 2 − eHn+ 3 2 h 2 L2_(Ω) + Jh t_n+3 2 − eJn+ 3 2 h 2 L2_(Ω) !1 2 , whereEen+1_h , eHn+ 3 2 h andeJ n+3 2 h are defined as :                                Z Ti e Hn+ 3 2 h −Hh tn+1 2 ∆t · ϕh = − Z Ti Eh(tn+1)· (∇ × ϕh) + X k∈Vi Z aik ϕh· ({Eh(tn+1)}_ik×nik) , ε∞ Z Ti e En+1_h ₋Eh(tn) ∆t · ψh = Z Ti Hh tn+1 2 · (∇ × ψh)− Z Ti J_htn+1 2 · ψh −X k∈Vi Z aik ψh× n Hh t_n+1 2 o ik ·nik, 3

Theoretical and numerical analysis of local dispersion models coupled to a discontinuous Galerkin time-domain method for Maxwell's equations

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https://hal.inria.fr/hal-00819758v2

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Theoretical and numerical analysis of local dispersion

models coupled to a discontinuous Galerkin time-domain

method for Maxwell’s equations

Jonathan Viquerat, Maciej Klemm, Stéphane Lanteri, Claire Scheid

To cite this version:

Jonathan Viquerat, Maciej Klemm, Stéphane Lanteri, Claire Scheid. Theoretical and numerical

anal-ysis of local dispersion models coupled to a discontinuous Galerkin time-domain method for Maxwell’s

equations. [Research Report] RR-8298, INRIA. 2013, pp.79. �hal-00819758v2�

RESEARCH

REPORT

N° 8298

numerical analysis of

local dispersion models

coupled to a

discontinuous Galerkin

time-domain method for

Maxwell’s equations

discontinuous Galerkin time-domain method

for Maxwell’s equations

Jonathan Viquerat, Maciej Klemm

, Stéphane Lanteri,

Claire Scheid

Theoretical and numerical analysis of local dispersion

models coupled to a discontinuous Galerkin time-domain

method for Maxwell’s equations

Contents

List of Figures

List of Tables

1 Photonics and dispersive media

1.1 Physical dispersion

1.2 Drude model

1.3 Drude-Lorentz model

1.4 Generalized dispersive model

1.5 Summary

2 DGTD method for non-dispersive media

2.1 Weak formulation

2.2 Space discretization

2.3 Time discretization

3 DGTD formulation in dispersive media

3.1 Previous works

3.2 DGTD method in Drude-like media

3.3 DGTD method for the generalized dispersive model

4 Theoretical study of the Maxwell-Drude equations

4.1 Stability study of the Maxwell-Drude equations

4.2 Convergence of the fully discrete Maxwell-Drude DG formulation

_{, Stéphane Lanteri,}