A NONDIFFUSIVE FINITE VOLUME SCHEME FOR THE ... - CiteSeerX

SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 2089–2108

c 2002 Society for Industrial and Applied Mathematics 

A NONDIFFUSIVE FINITE VOLUME SCHEME FOR THE THREE-DIMENSIONAL MAXWELL’S EQUATIONS ON UNSTRUCTURED MESHES∗ SERGE PIPERNO† , MALIKA REMAKI† , AND LOULA FEZOUI† Abstract. We prove a sufficient CFL-like condition for the L2 -stability of the second-order accurate finite volume scheme proposed by Remaki for the time-domain solution of Maxwell’s equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions. Numerical tests show the potential of this original finite volume scheme in one, two, and three space dimensions for the numerical solution of Maxwell’s equations in the time-domain. Key words. electromagnetics, finite volume methods, centered fluxes, leap-frog time scheme, L2 -stability, energy methods, unstructured meshes, absorbing boundary condition AMS subject classifications. 65M12, 65M60, 76M12, 78A45, 78A50, 78M25 PII. S0036142901387683

1. Introduction. The modeling of systems involving electromagnetic waves has known a kind of reinvention [28] through the resolution of the time-domain Maxwell’s equations on space grids. Many different types of methods have been used. Finite difference time-domain (FDTD) methods (based on Yee’s scheme [34] or on implicit time schemes [25]) are efficient mostly on structured regular grids, whereas finite element methods, based on unstructured meshes, can deal with naturally complex geometries [8] but induce heavy computations of mass matrices. Although some significant work has been devoted to the construction of edge elements allowing accurate and efficient mass lumping [11, 12], actual numerical results on unstructured grids for transient problems are very recent [17]. Besides the finite elements methods, we can cite the emerging mimetic methods which also aim at dealing with nonuniform grids. They have been applied recently to Maxwell’s system [15]. The proven properties of these methods make them close to the classical edge elements in the unstructured case. Gathering many advantages, finite volume time-domain (FVTD) methods can also be based on unstructured meshes and get rid of differential operators (and finite element mass matrices) using Green’s formula for the integration over finite volumes. (See [9] for a review of numerical methods used in computational electromagnetics and [29] for an accurate review of FDTD and FVTD methods.) We are interested here in FVTD methods, as have been developed in the past years, not necessarily on body-fitted coordinates [26, 27] but on unstructured finite element triangulations [6, 7, 8, 24] or on totally destructured meshes [3]. More precisely, we consider a standard finite volume approximation, i.e., a piecewise constant, discontinuous, Galerkin-type finite element approximation [18]. As the Maxwell’s system in transient state is hyperbolic and may be rewritten in conservative form, it is natural to use a numerical approximation based on conservative schemes in a first step directly inspired by previous works in the field of computational fluid dynam∗ Received by the editors April 10, 2001; accepted for publication (in revised form) October 23, 2001; published electronically April 3, 2002. http://www.siam.org/journals/sinum/39-6/38768.html † INRIA, BP93, 06902 Sophia Antipolis Cedex, France ([email protected], Malika. [email protected], [email protected]).

2089

2090

SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

ics. The convergence of first-order conservative upwind schemes has been established for different hyperbolic equations in any dimension [13], and L1 error estimates of h1/2 (where h is a characteristic mesh size) have been proven recently for a general hyperbolic equation [31]. The stability of finite volume schemes has been investigated for many years for regular grids using the Von Neumann analysis [1] or the modified equation analysis [33]. Both analyses do not deal either with boundary conditions or with nonregular grids. The concept of total variation diminishing (TVD) scheme, proposed by Harten [14], leads to L∞ -stability results for finite volume schemes on nonregular grids only in one space dimension and can be related to the idea of local extremum diminishing (LED) scheme, proposed by Jameson [16], which extends some properties to several dimensions. In a previous paper [21], we have investigated the L2 -stability of first-order upwind finite volume schemes in two and three space dimensions on unstructured meshes for the numerical solution of the time-domain Maxwell’s equations. Quasi-optimal sufficient stability conditions were obtained on arbitrary finite volume partitions, which were mentioned and actually used [3] and which extend a general condition derived for Friedrichs’ systems in general [32]. However, upwind schemes are very disappointing when used for the numerical simulation of electromagnetic waves propagation, since the numerical diffusion induced by upwinding—necessary in computational fluid dynamics to limit unphysical oscillations—makes long-run computations (at least over several periods) very inaccurate. In this paper, we investigate a new finite volume method proposed by Remaki [23] for the numerical solution of Maxwell’s equations in heterogeneous media. In that heterogeneous framework, a result of existence and uniqueness of the solution is now available [22]. Remaki’s FVTD method is based on a second-order leap-frog time scheme and on second-order centered numerical fluxes. This scheme yields an original conservative finite volume method with no numerical diffusion (in a sense to be discussed later). Finite volumes of arbitrary shape are considered, as well as two types of boundary conditions (absorbing and metallic boundary conditions). As done previously for upwind schemes on arbitrary meshes, an energy-type method, drawn from some finite element proofs [5], is used to prove the L2 -stability of the scheme. Concerning accuracy and convergence, it is easily proven that the finite volume is actually second-order accurate on regular orthogonal grids (because it can be seen as a finite difference method). However, we do not address in this paper the question of convergence and error estimates in the general case of nonuniform meshes. This paper is organized has follows. In section 2, we introduce a quasi-conservative form of Maxwell’s equations in three dimensions for heterogeneous media; we also introduce some notations. In section 3, we recall the centered leap-frog second-order finite volume scheme proposed by Remaki for the time-domain solution of Maxwell’s equations. In section 4, we prove the sufficient stability condition for this scheme. This proof is based on the definition of a discrete electromagnetic energy. We also prove in this section that this energy is exactly conserved if no absorbing boundaries are present. Finally, we present in section 5 some numerical results in two and three dimensions, both for homogeneous and heterogeneous media. 2. The heterogeneous three-dimensional Maxwell’s equations. We consider Maxwell’s equations in three space dimensions (heterogeneous linear isotropic medium with no source, with space varying electric permittivity (x) and magnetic permeability µ(x), the local light speed c(x) being given by (x)µ(x)c(x)2 = 1). The t t electric field E = (Ex , Ey , Ez ) and the magnetic field H = (Hx , Hy , Hz ) verify

2091

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD δΩm ⇔ δΩ a

8

δΩ

δΩm

Ω

Ω

RCS of a wing

Cavity resonance

Fig. 2.1. Domain Ω and domain boundary.

 ∂E     ∂t = ∇ × H,    µ ∂H = −∇ × E. ∂t These equations are set in a bounded polyhedral domain Ω of R3 . Everywhere on the domain boundary ∂Ω, exactly one of the two possible boundary conditions is set: a metallic boundary condition (on ∂Ωm , around a metallic object, or inside a cavity, for example) or an absorbing boundary condition (on ∂Ωa , possibly on the outer boundary of the domain ∂Ω∞ ; see Figure 2.1). In three space dimensions, the metallic boundary condition is written as n×E = 0, with n the unitary outwards normal. In the following, a first-order Silver–M¨ uller absorbing condition is used on the absorbing boundary ∂Ωa , which is written as

n × E = −cµ n × ( n × H) . The Maxwell’s system can be transformed into a pseudoconservative form:  ∂E ∂H ∂H ∂H     ∂t + Nx ∂x + Ny ∂y + Nz ∂z = 0, (2.1)    µ ∂H − Nx ∂E − Ny ∂E − Nz ∂E = 0, ∂t ∂x ∂y ∂z where the antisymmetric matrices Nx , Ny ,    0 0 0 0 N x =  0 0 1  , Ny =  0 0 −1 0 1

and Nz are given by   0 −1 0 0 0  , Nz =  −1 0 0 0

1 0 0

 0 0 . 0

For any vector n = t (nx , ny , nz ), the matrix Nn = nx Nx + ny Ny + nz Nz is antisymmetric and operates as Nn X = − n × X. 3. The centered leap-frog second-order finite volume scheme. 3.1. Introduction. We assume we dispose of an arbitrary partition of the domain Ω into a finite number of connected polyhedral finite volumes (each one with a finite number of faces). For example, this assumption covers the two cases of vertexcentered and element-centered finite volumes [7]. For each finite volume or “cell” Ti ,

2092

SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

Vi denotes its volume, and i , µi , and zi = µi /i are, respectively, the local electric permittivity, magnetic permeability, and impedance of the medium, which are assumed constant in the cell Ti . We call the interface between two finite volumes their intersection whenever it

is a polyhedral surface. For each internal interface aij = Ti Tj , we denote by t

nij = (nijx , nijy , nijz ) the integral over the interface of the unitary normal, oriented from Ti towards Tj . The same definitions are  extended to boundary interfaces (in the intersection of the domain boundary ∂Ωm ∂Ωa with a boundary finite volume), the index j corresponding to a fictitious cell outside the domain. We denote t by n ˜ ij = (˜ nijx , n ˜ ijy , n ˜ ijz ) the normalized normals n ˜ ij = nij / nij . For the sake of simplicity, we use the notation Nij = Nnij =  nij  Nn˜ ij . Finally, we denote by Vi the set of indices of the neighboring finite volumes of the finite volume Ti (having an interface in common). We denote by Si the discrete measure of the boundary of a finite volume cell, which is defined by  nij . Si = j∈Vi

3.2. Conservative finite volume scheme. The scheme proposed by Remaki [23] is written  n+1/2 Ein+1 − Ein    + V Fij = 0, i i   ∆t  j∈Vi (3.1) n+3/2  − Hin+1/2   µi Vi Hi  − Gn+1 = 0, ij  ∆t j∈Vi

where the index i is linked to the cell Ti , ∆t is the time step, Ein (respectively, Hin+1/2 ) is an approximate for the average of the electric (respectively, magnetic) field over the cell Ti at time tn = n ∆t (respectively, tn+1/2 = (n + 12 ) ∆t). The word average has to be understood as a space-mean value, which is defined for any field X and each cell Ti by

1 X(s) ds. Xi = Vi Ti Finally, Fijn+1/2 and Gn+1 are centered numerical fluxes. The reader can check that ij the finite volume scheme (3.1) is written as a leap-frog scheme: numerical values Ein and Hin+1/2 are used in numerical fluxes to obtain a new set of numerical values Ein+1 and Hin+3/2 . In what follows, we use the very important notation Gnij + Gn+1 Ein + Ein+1 ij n+1/2 and Gij ≡ ≡ (3.2) . 2 2 The numerical fluxes for each kind of interfaces (sketched on Figure 3.1) are defined as follows: are given for internal interfaces by • The fluxes Fijn+1/2 and Gn+1 ij Ein+1/2

(3.3)

(3.4)

Hin+1/2 + Hjn+1/2 Ein+1 + Ejn+1 n+1 , Gij . = Nij 2 2 • For a metallic boundary interface, we use the following numerical fluxes: Fijn+1/2 = Nij

Fijn+1/2 = Nij Hin+1/2 ,

Gn+1 = 0, ij

2093

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD

Metallic interfaces

Internal interfaces

Absorbing interfaces

Fig. 3.1. Different fluxes for different kinds of interfaces.

which could be obtained using a fictitious cell with Hjn+1/2 = Hin+1/2 and Ejn+1 = −Ein+1 . The choice for Gn+1 is clearly a second-order approximation ij of the condition n × E = 0 at the interface, whereas the choice for Fijn+1/2 is classically used in FDTD methods. For Maxwell’s equations in one dimension, this weak treatment leads to an accurate reflection of incoming waves. • Finally, for an interface aij on the absorbing boundary Ωa , the numerical will be detailed in what follows. fluxes Fijn+1/2 and Gn+1 ij Remark 3.1. Centered numerical fluxes and heterogeneous materials. The definition of the numerical fluxes (3.3) apparently ignores the fact that the electric and magnetic fields can be discontinuous on a material interface (for instance, the interface between two homogeneous media). Anyway, at such a material discontinuity, only the normal parts of the fields should be discontinuous, and these normal parts vanish in the expressions (3.3) of the numerical fluxes because of the definition of Nij . 3.3. Matricial properties. The following elementary equalities hold:

nji = − nij ,

Nji = −Nij ,

interfaces of Ti

nij = 0,



Nij = 0.

interfaces of Ti

4. A sufficient stability condition. 4.1. Energy estimates. We aim at giving and proving a necessary and/or sufficient condition for the L2 -stability of the finite volume scheme (3.1), (3.3), (3.4) with absorbing boundary fluxes to be determined. We use the same kind of energy approach as in [21], where a quadratic form plays the role of a Lyapunov function of the whole set of numerical unknowns. We first propose the following discrete energy, directly derived from the expression of the total numerical electromagnetic energy: (4.1)

En =

 1  t n n t Vi i Ei Ei + µi Hin−1/2 Hin+1/2 . 2 i

We recall here that the electromagnetic energy in the continuous case verifies some conservation equation (Poynting’s theorem) for the Maxwell’s system with no current. This theorem states that   ∂ E dx + (4.2) P · nds = 0 ∂t V ∂V

2094

SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

for any closed volume V with a regular boundary ∂V , where the electromagnetic energy E and Poynting’s vector P are, respectively, given by 1 1 E = εE 2 + µH 2 , (4.3) P = E × H. 2 2 For a given metallic cavity, since E × n = 0 at the boundary, Poynting’s theorem yields that the electromagnetic energy is exactly conserved in the cavity. Following this idea, we naturally try to use the proposed discrete energy (4.1) as a Lyapunov function. It is not absolutely obvious why the discrete energy (4.1) should be a positive definite quadratic form of all numerical unknowns (let us say the Ein and the Hin−1/2 ). We notice here that the situation is quite different from the proof of the L2 -stability of the first-order upwind finite volume scheme of [21], where the energy was obviously a positive definite quadratic form of all unknowns. At the same time, the energy proposed here depends explicitly on the numerical scheme, since it can be only written as a quadratic form of all unknowns (Ein , Hin−1/2 ) through the use of the second part of the scheme (3.1). Finally, the variation of this discrete energy during a time step might lead to tedious computations, as for the first-order upwind scheme. In the following, we shall prove that the proposed energy with additional boundary correction terms is both nonincreasing during a time step and a positive definite quadratic form of all unknowns under a CFL-like condition on the time step ∆t. This will yield the proof that the scheme is L2 -stable (with the proposed energy) under a stability condition on ∆t. We define the energy variation ∆E = E n+1 −E n and propose the following lemma. Lemma 4.1. Using the scheme (3.1), (3.3), (3.4) with some absorbing boundary fluxes Fijn+1/2 and Gn+1 (to be defined), we have the following estimation for ∆E: ij ∆E = ∆t

absorbing interfaces

 t n+1/2  n+1/2   − Ei Fij − Nij Hin+1/2 + tHin+1/2 Gnij+1/2 .

Proof. We have   1 t n ∆E = Ei + Ein+1 i Vi (Ein+1 −Ein ) 2 i   + tHin+1/2 µi Vi (Hin+3/2 −Hin+1/2 ) + µi Vi (Hin+1/2 −Hin−1/2 )   t = ∆t −tEin+1/2 Fijn+1/2 + Hin+1/2 Gnij+1/2 . i

j∈Vi

All terms in the double summation above correspond to finite volume interfaces. These terms can then be distributed on volume interfaces. We have   ∆E = ∆t . Tinternal + Tmetallic + Tabsorbing , with Tinternal = Tmetallic = Tabsorbing =

internal



interfaces metallic 

 t t t t n+1/2 −Ein+1/2 Fijn+1/2 − Ejn+1/2 Fji + Hin+1/2 Gnij+1/2 + Hjn+1/2 Gnji+1/2 ,  t t −Ein+1/2 Fijn+1/2 + Hin+1/2 Gnij+1/2 ,

interfaces absorbing  interfaces

 t t −Ein+1/2 Fijn+1/2 + Hin+1/2 Gnij+1/2 .

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD

2095

For the internal interface term, we can replace the centered numerical fluxes by the formula (3.3). Using the identities t Nij = −Nij = Nji , we find 

internal

Tinternal =

interfaces

t n+1/2 Hi Nij Ein+1/2

 t + Hjn+1/2 Nji Ejn+1/2 .

For the metallic interface term, we can replace the metallic fluxes by the formula (3.4). We easily find that Tmetallic =

metallic interfaces



t n+1/2 Hi Nij Ein+1/2



.

Finally, we find that many terms vanish, since absorbing  interfaces

=

t n+1/2 Hi Nij Ein+1/2

internal



interfaces

+



+ Tmetallic + Tinternal

t n+1/2 Hi Nij Ein+1/2



metallic interfaces

t

+ Hjn+1/2 Nji Ejn+1/2

t n+1/2 Hi Nij Ein+1/2



+

absorbing  interfaces

t n+1/2 Hi Nij Ein+1/2



    t n+1/2 t  Hin+1/2  Hi Nij Ein+1/2  = Nij  Ein+1/2  = 0. = i







i

j∈Vi

j∈Vi

We then have ∆E = ∆t

absorbing  interfaces

   t t −Ein+1/2 Fijn+1/2 − Nij Hin+1/2 + Hin+1/2 Gnij+1/2 ,

which is exactly the result of the lemma. Remark 4.1. Global conservation of the energy with no absorbing boundary. The lemma above shows that the energy variation is only due to absorbing boundaries. One consequence is that with only metallic boundaries, the discrete energy is globally exactly conserved. Remark 4.2. Local conservation of the energy. A slightly different version of Lemma 4.1 can be simply derived for any group G of connected tetrahedra (we denote by aij the interfaces of the boundary ∂G, i and j being, respectively, related to inside and outside vertices). If EGn denotes the discrete electromagnetic energy inside the group G at time tn , (4.4)

EGn =

 1  t n n t Vi i Ei Ei + µi Hin−1/2 Hin+1/2 , 2 i∈G

we can show that the variation of this local energy over one time step is given by

2096

SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

∆EG =

EGn+1

−

EGn

= −∆t

aij ∈∂G

t

 Ein+1/2 × Hjn+1/2 + Ejn+1/2 × Hin+1/2 · nij . 2

This equation is exactly a discrete version of Poynting’s theorem (4.2)–(4.3). This shows that the discrete electromagnetic energy verifies a local discrete conservation law as in the continuous equations, even though the electromagnetic energy is not a conservative variable of the hyperbolic system of Maxwell’s equations. Remark 4.3. Nondiffusivity. We claim that the FVTD method is actually nondiffusive. On one hand, one can easily check (at least for the Maxwell’s equations in one space dimension) that the modified equation [33] for the proposed finite volume method on a regular grid contains no diffusive terms. On the other hand, we can also notice that, if these terms were present, they would lead to the nonconservation of any quadratic positive definite form of the unknowns, like the discrete electromagnetic energy. Since some discrete energy is exactly conserved even for unstructured grids, our claim is justified in the general case. If some absorbing boundary conditions are present, in view of the result of the preceding lemma, we can now propose some choices for the absorbing boundary fluxes Fijn+1/2 and Gn+1 ij :   nij  n Nij n+1/2 n+1/2   Hi Ei Tij , = + zi −1  Fij 2 2 (4.5)    Gn+1 = Nij E n+1 − z  nij  H n+1/2 , i i Tij ij 2 i 2 ˜ ij .X) n ˜ ij . where XTij denotes the tangential part of a vector X, given by XTij = X −(t n n+1/2 n In these definitions, Ei Tij and Hi Tij are, respectively, the tangential (orthogonal to the normal nij ) parts of the vectors Ein and Hin+1/2 . The reader can also notice that the fields Ein and Hin+1/2 are available when the boundary flux Fijn+1/2 is used and then that the fields Ein+1 and Hin+1/2 are also available when the boundary flux Gn+1 is used. ij The origin of these fluxes might not seem really obvious. In fact, they correspond to upwind fluxes at the absorbing boundary. More precisely, the reader can check that the flux Fijn+1/2 proposed above corresponds  n n+1/2to  the three electric field components of the +t E i , Hi six-component upwind flux Mij , where Mij = Ax nij x +Ay nij y +Az nij z , the matrices Ax , Ay , Az correspond to the operators in the conservative form of the Maxwell’s equations in function of the complete electromagnetic field t(E, H), and the superscript + corresponds to the positive part via diagonalization of a matrix (see [21] for complete details). Similarly, Gn+1 corresponds to the ij  n+1 n+1  three magnetic components +t Ei , Hi /2 . The reader can notice that of the six-component upwind flux Mij these fluxes lead to a genuinely first-order treatment of the absorbing condition, since the approximation is first order both in space (upwind fluxes) and in time. Fijn+1/2 is based on Ein and Gn+1 on Hin+1/2 . With some time-interpolation, these fluxes can ij be corrected to reach second-order accuracy for Maxwell’s equations in one dimension. We plan to compare these treatments of the absorbing condition with Berenger’s perfectly matched layers (PML) [4], which are largely used in computational electromagnetics simulations [2, 10, 20, 19]. However, our main goal here is to get a stability result, and we are actually able to reach that goal only for these first-order accurate absorbing fluxes, as will be shown later in the paper. With the choice (4.5) for the boundary fluxes, a value for the variation ∆E of the discrete energy (4.1) deriving from Lemma 4.1 is given in the following lemma.

2097

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD

Lemma 4.2. Using the scheme (3.1), (3.3), (3.4), (4.5), ∆E is given by

∆E = −

absorbing

  t n+1/2 n−1/2 n+1/2  nij  zi −1 tEi nTij (Ei nTij + Ei n+1 ) + z H (H + H ) . i i i i Tij Tij Tij Tij

∆t 4

interfaces

Proof. Lemma 4.1 yields

∆E = ∆t

absorbing  interfaces

=

∆t 2

   t t −Ein+1/2 Fijn+1/2 − Nij Hin+1/2 + Hin+1/2 Gnij+1/2

absorbing 

t

interfaces

+ =−

∆t 4

absorbing interfaces

Ein + Ein+1 t

 Nij n+1/2 t n   nij  n H − Ei + Ein+1 Ei Tij 2 i 2zi

Nij Hin+1/2



2

Ein

+

Ein+1



 zi  nij  t n+1/2 n−1/2 n+1/2 − Hi (Hi Tij + Hi Tij ) 2

  t n+1/2 n−1/2 n+1/2  nij  zi −1 tEi nTij (Ei nTij + Ei n+1 ) + z H (H + H ) , i i i i Tij Tij Tij Tij

which is the result of the lemma. 4.2. A corrected discrete energy. The discrete energy E n proposed in (4.1) depends explicitly on the “updating scheme” for the magnetic field, and therefore on the particular choice for Gn+1 ij . Then it is not really surprising that this energy will have to be corrected to help us in our proof. Let us then introduce a corrected discrete energy F n , given by

(4.6)

F n = E n − ∆t

absorbing interfaces

  nij   −1 2 −1/2 2 zi Ei nTij  − zi Hi Tnij  . 8

The physical meaning of this corrected discrete energy is not straightforward. Correction terms are only related to absorbing boundaries (which means that F n = E n if there are none). The additional terms probably find their origin in the temporal inconsistency of boundary numerical fluxes. We can now prove that the discrete energy F n is nonincreasing when our centered finite volume scheme is used. This is summed up in the following lemma. Lemma 4.3. Using the scheme (3.1), (3.3), (3.4), (4.5), the discrete energy F n given by (4.1)–(4.6) is nonincreasing: the variation ∆F = F n+1 − F n is given by

∆t ∆F = − 2

absorbing



 nij  zi −1

interfaces

Ei nTij + Ei n+1 Tij 2

2

+ zi

−1/2 +1/2 Hi Tnij + Hi Tnij

2

2

  ≤ 0.

2098

SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

Proof. The proof is elementary. We simply add terms which derive from the correction in F n to the result of Lemma 4.2 for ∆E. We have ∆t ∆F = − 2

absorbing

  nij  zi

−1 t

interfaces

− ∆t

Ei nTij

absorbing interfaces

+ ∆t

absorbing interfaces

∆t =− 2

absorbing



 nij zi −1

interfaces

Ei nTij + Ei n+1 Tij 2

+

t

+1/2 zi Hi Tnij

−1/2 +1/2 Hi Tnij + Hi nTij



2

  nij   −1 2 n+1/2 2 zi Ei Tn+1  − z H  i i Tij ij 8   nij   −1 2 −1/2 2 zi Ei nTij  − zi Hi Tnij  8 Ei nTij + Ei n+1 Tij 2

2

+ zi

−1/2 +1/2 Hi Tnij + Hi nTij

2

2

 .

We have proven that the scheme (3.1), (3.3), (3.4), (4.5) is such that the discrete energy F n is nonincreasing, hence it is bounded by its initial value. We can notice here that this result is valid independently of the value of the time step ∆t. 4.3. A sufficient stability condition. In order to prove that our scheme is stable, we finally show that the discrete energy F n , under a stability condition on ∆t, is a positive definite quadratic form of the numerical values Hin−1/2 and Ein . This leads to the result of this paper, taking the form of the following stability. Theorem 4.4. Using the scheme (3.1), (3.3), (3.4), (4.5) on arbitrary polyhedral finite volumes as described in this section, the energy F n defined in (4.6) is a nonincreasing, positive definite quadratic form of all unknowns (Ein , Hin−1/2 ), and therefore the scheme is L2 -stable if the time step ∆t is such that ∀ interface aij , ∆t2 < 16

Vi Vj min(j µi , i µj ). Si Sj

In the above formula, j should be replaced by i for boundary interfaces aij . Proof. We return to the definition of the discrete energy F n . We have Fn =

 1  t n n t Vi i Ei Ei + µi Hin−1/2 Hin+1/2 2 i −

∆t 2

absorbing interfaces

  nij   −1 2 −1/2 2 zi Ei nTij  − zi Hi Tnij  . 4

The magnetic field Hin+1/2 is given by (3.1) and depends linearly on Hin−1/2 and the fluxes Gnij . The latter depend linearly in all cases on the Ein and for the absorbing boundary on the Hin−1/2 through (4.5). Then it is clear that F n and E n are quadratic forms of the unknowns (Ein , Hin−1/2 ). We now need a lower bound for F n . We have

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A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD

2E n =

i

=



  t Vi i tEin Ein + µi Hin−1/2 Hin+1/2  2

Vi i Ein 2

+ µi Vi Hin−1/2  + ∆t

i

=

=



 Gnij 

j∈Vi

i

t n−1/2 Hi

 Vi i Ein 2 + µi Vi Hin−1/2 2 + ∆t tHin−1/2



Gnij −

j∈Vi





Nij n  E 2 i

   Vi i µi Vi Nij n 2 t 2 Ei .  nij Ein  +  nij Hin−1/2  + ∆t Hin−1/2 Gnij − Si Si 2 i j∈Vi

In the above expression, the double summation over cells and cell neighbors can be redistributed over cell interfaces. (Internal interfaces will gather two contributions from neighboring cells, whereas boundary interfaces will gather only one inwards contribution.) We then have

2E n =

internal



interfaces

Vi i µi Vi ∆t t n−1/2 2 2  nij Ein  +  nij Hin−1/2  + Hi Nij Ejn Si Si 2

 Vj j µj Vj ∆t t n−1/2 2 2  nij Ejn  +  nij Hjn−1/2  − Hj Nij Ein Sj Sj 2   metallic Vi i µi Vi ∆t t n−1/2 2 2 +  nij Ein  +  nij Hin−1/2  − Hi Nij Ein Si Si 2 +

interfaces

+

absorbing  interfaces

Vi i µi Vi ∆t 2 2 −1/2  nij zi Hi Tnij  nij Ein  +  nij Hin−1/2  − Si Si 2

2

 .

Using elementary lower bounds for scalar products, the definition of the matrix Nij , and adding correction terms in F n , we find that

n

2F ≥

 

internal

 nij 

interfaces

 +

+

metallic



 nij 

interfaces

+

absorbing





∆t Vi i n 2 µj Vj 2 Hjn−1/2 Ein  Ei  + Hjn−1/2  − Si Sj 2 

Vi i n 2 µi Vi ∆t n 2 Ei Hin−1/2  Ei  + Hin−1/2  − Si Si 2

Vi i n 2 µi Vi ∆t 2  nij  Ei  + Hin−1/2  − Si Si 4

interfaces



Vj j µi Vi ∆t 2 2 Ejn  + Hin−1/2  − Hin−1/2 Ejn  Sj Si 2



−1/2 zi Hi Tnij

2

1 Ei nTij + zi

2

 .

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SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

Thus, if all discriminants are strictly negative, i.e., the time step ∆t is such that  V V (i) ∀ internal interface aij , ∆t2 < 16 Sii Sjj min(j µi , i µj ),    (ii) ∀ metallic interface near cell Ci , ∆t < 4 cVi Si i ,    (iii) ∀ absorbing interface near cell Ci , ∆t < 4 cVi Si i , it is clear that F n ≥ 0 and that F n = 0 ⇒ (∀ i, Ein = 0, Hin−1/2 = 0). This concludes the proof that under the condition of Theorem 4.4, the discrete energy F n is a positive definite quadratic form of all numerical unknowns. It is also nonincreasing, then bounded, and the scheme is L2 -stable. Remark 4.4. Comparison with Yee’s scheme. The comparison of the sufficient stability condition given in Theorem 4.4 with the stability limit obtained by a Von Neumann analysis of Fourier eigenmodes for a structured orthogonal regular grid yields an estimation of the optimality of this sufficient condition. For an orthogonal Cartesian grid made of square hexahedra ∆x×∆y ×∆z and for an homogeneous medium, it can be checked that the centered finite volume scheme coincides with eight Yee’s schemes [34] on eight three-dimensional staggered grids of meshsteps (2∆x, 2∆y, 2∆z); hence, it is stable if and only if ! 1 1 1 c∆t + + ≤2 2 2 ∆x ∆y ∆z 2 (see [30] for a correct proof of the stability limit of Yee’s scheme). For the same Cartesian grid, our general sufficient stability condition is   1 1 1 c∆t + ≤ 2, + ∆x ∆y ∆z which is more restrictive, but note that it is only sufficient and not necessary. Although the new FVTD scheme with a twice finer grid recovers the same stability limit and dispersion as Yee’s scheme, the main property of this scheme is its ability to handle complex geometries using unstructured grids. It does not appear as an unstructured generalization of Yee’s scheme, because unknowns of both electric and magnetic fields are collocated and represent average values over control volumic cells, which is standard in finite volume methods. In what follows, some numerical results show the basic property of the scheme, including on unstructured grids around complex geometries. 5. Numerical results. 5.1. A rectangular waveguide with a current source (two dimensions). We consider a current jz over the √ cross section of a rectangular waveguide at x = 0 (see Figure 5.1), given by jz (y) = 2 sin(πy/d) cos(2πf t), where d = 1m is the waveguide width and f = 0.3Ghz the signal frequency. The grid used here corresponds to 12 points per wavelength. Figure 5.2 and Figure 5.3 represent the electric and the magnetic fields produced by the current jz . We can see on Figure 5.2 the good quality of the approximate solutions, which compare very well with exact solutions. 5.2. Scattered waves across a dielectric coated airfoil (two dimensions). We simulate scattering problems across airfoils in heterogeneous cases. We first illuminate a coated NACA0012 airfoil profile (dielectric layer of thickness δ = 0.1λ,

2101

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD metallic boundary

y

d

jz metallic boundary

x Fig. 5.1. A current sheet in a rectangular waveguide.

1 EXACT SOLUTION APPROXIMATE SOLUTION 0.8 0.6 0.4

Ez field

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -6

-4

-2

0 X axis

2

4

6

0.8 EXACT SOLUTION APPROXIMATE SOLUTION 0.6

0.4

Hy field

0.2

0

-0.2

-0.4

-0.6

-0.8 -6

-4

-2

0 X axis

2

4

6

Fig. 5.2. Section at y= d2 , Ez (above) and Hy (below). Numerical and exact solutions.

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SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

Fig. 5.3. Numerical Ez (above) and Hy (below).

Fig. 5.4. The unstructured discretization for the coated NACA0012 airfoil. (The dielectric material corresponds to triangles inside the marked line.)

with ε = 4 and µ=1) by a monochromatic wave located rightwards (with frequency f = 1.2Ghz). The space discretization is unstructured and corresponds to 15 points per wavelength (see Figure 5.4). Figure 5.5 represents the contours of the scattered electric field. We can notice that the numerical solution has no spurious oscillation at the interface between the two materials. 5.3. Scattered waves across a coated cylinder (two dimensions). We present here as an example the computed result for the magnetic field scattered by a model antenna (here an heterogeneous circular dielectric cylinder scatterer), which is

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD

2103

Fig. 5.5. The scattered electric field.

Free space

Dielectric Metal

Fig. 5.6. Geometry (left), zoom of the mesh (right).

pictured on Figure 5.6. The frequency of the incident wave coming from the right side of the object is f = 0.15Ghz. The radius of the large metallic cylinder is R = 0.459m, while the radius of small metallic cylindrical inclusions is R = 0.025m (the centers are regularly spaced with an interval of 0.1m), and the thickness of the dielectric layer is δ = 0.1m. The perfect dielectric material is defined by εr = 2.56 and µr =1. In this test case, the mesh is rather coarse, since we have only 14 points per wavelength. Again, the scattered magnetic field contoured on Figure 5.7 presents no spurious oscillations at the material interface.

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SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

Fig. 5.7. Scattered magnetic field.

5.4. Electromagnetic compatibility. We consider a window in an airfoil with a slit as in Figure 5.8. The relative parameters of the dielectric layer are εr = 2 and µr = 1. We consider different imperfectly conducting materials (different σ). We send a monochromatic wave with incidence angle θ = 90o , and frequency f = 0.3 Ghz. We compute the wave propagation across the material (Figures 5.9 and 5.10). These simulations were produced with an exponential time scheme for the treatment of the additional loss terms. Depending on the conductivity σ of the nonperfect conductor, we get different total electric field with a smooth transition from an apparent perfect dielectric (σ = 0) to an almost symmetric solution (for σ = 100, we have almost two perfect conductors). 5.5. Eigenmodes in a three-dimensional metallic cavity. We present here preliminary results for the numerical simulation of the time evolution of an eigenmode in a cubic metallic cavity. We consider the (1,1,1) mode. The cubic cavity is discretized by a structured orthogonal regular cube, with twenty cells in each direction. We have chosen to present on Figure 5.11 the vertical component Hz of the magnetic field at the center of the cavity in function of the time. Numerical results obtained by Yee’s scheme and our finite volume scheme (with cubic elementary volumes) are comparable: no numerical dissipation is artificially produced, and the L2 error with the exact solution (pointwise differences for both grids) is very small. For each numerical scheme, we have plotted the exact and approximate value correspond-

A NONDIFFUSIVE UNSTRUCTURED FVTD METHOD

2105

wave incident

a dielectric layer (a slit) ( ε = 2)

a perfect conductor

a conductor (σ >> 1)

Fig. 5.8. A window’s slit.

Fig. 5.9. Total electric field: σ = 0 (left) and σ = 100 (right).

Fig. 5.10. Total magnetic field: σ = 0 (left) and σ = 100 (right).

ing to the considered degree of freedom. Since Yee’s scheme involves a staggered grid, both “exact” curves differ. We can observe that, on this particular case of a regular orthogonal grid, the numerical dispersions and the computational costs of both FVTD and FDTD methods are comparable.

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SERGE PIPERNO, MALIKA REMAKI, AND LOULA FEZOUI

0.25 Hz FVTD Hz FDTD exact (fvtd) exact(fdtd)

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0

2

4

6

8

10

12

14

Fig. 5.11. Vertical magnetic field at the center of a metallic cubic cavity for Yee’s scheme and our finite volume scheme.

6. Conclusion. In this paper, we have proposed a sufficient condition for the stability of the second-order finite volume scheme proposed by Remaki for the timedomain solution of Maxwell’s equations in two and three dimensions in heterogeneous media. Energy estimates lead us to a sufficient CFL-like stability conditions on arbitrary finite volumes with metallic or absorbing boundary conditions, for which weak treatments with at least consistent numerical fluxes have been used. The stability condition happens to be more restrictive—but with a comparable limit time step—on regular grids than that obtained with Fourier analysis. However, this condition is general. It is valid in three space dimensions on arbitrary polyhedral finite volumes. Some numerical results in two and three dimensions prove the large potential of the method, for which some work still lies ahead: proof of convergence and second-order accuracy on unstructured grids (possibly of other centered schemes), coupling with PML medium, and multiscale coupling of multiple subdomains with different mesh sizes and time steps. REFERENCES [1] D.A. Anderson, J.C. Tannehill, and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere, Washington, DC, McGraw-Hill, New York, 1984. [2] J.-D. Benamou and B. Despres, A domain decomposition method for the Helmholtz equation and related optimal control problems, J. Comput. Phys., 136 (1997), pp. 68–82. [3] F. Bourdel, P.-A. Mazet, and P. Helluy, Resolution of the non-stationary or harmonic Maxwell equations by a discontinuous finite element method. Application to an E.M.I. (electromagnetic impulse) case, in Computing Methods in Applied Sciences and Engineering, Nova Science, New York, 1991, pp. 405–422. [4] J.P. Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 127 (1996), pp. 363–379. [5] P.G. Ciarlet and J.-L. Lions, eds., Handbook of Numerical Analysis, Vol. 2, Part I, North

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mensional scalar conservation laws. I. Explicit monotone schemes, RAIRO Mod´ el. Math. Anal. Num´ er., 28 (1994), pp. 267–295. [32] J.-P. Vila and P. Villedieu, Convergence de la m´ ethode des volumes finis pour les syst` emes de Friedrichs, C. R. Acad. Sci. Paris S´er. I Math., 3 (1997), pp. 671–676. [33] R.F. Warming and F. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys., 14 (1974), pp. 159–179. [34] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas and Propagation, AP-16 (1966), pp. 302–307.