A High-Order Accurate Method for Frequency Domain Maxwell Equations with Discontinuous Coefficients

DOI: 10.1007/s10915-005-9049-5Journal of Scientific Computing, Vol. 27, Nos. 1–3, June 2006 (© 2006)

A High-Order Accurate Method for Frequency DomainMaxwell Equations with Discontinuous Coefficients

Eugene Kashdan1 and Eli Turkel1

Received September 26, 2004; accepted (in revised form) April 20, 2005; Published online March 13, 2006

Maxwell equations contain a dielectric coefficient ε that describes the particularmedia. For homogeneous materials the dielectric coefficient is constant. Thereis a jump in this coefficient across the interface between differing media. Thisdiscontinuity can significantly reduce the order of accuracy of the numericalscheme. We present an analysis and implementation of a fourth order accu-rate algorithm for the solution of Maxwell equations with an interface betweentwo media and so the dielectric coefficient is discontinuous. We approximatethe discontinuous function by a continuous one either locally or in the entiredomain. We study the one-dimensional system in frequency space. We onlyconsider schemes that can be implemented for multidimensional problems bothin the frequency and time domains.

KEY WORDS: Maxwell equations; Helmholtz equation; discontinuous coeffi-cients; high-order method.

1. INTRODUCTION

The Maxwell equations for �E, �D, �H and �B are:

∂ �B∂t

+∇ × �E =0, (Faraday’s Law)

(1)∂ �D∂t

−∇ × �H =− �J , (Ampere’s law)

coupled with Gauss’s law

∇ · �B =0, ∇ · �D =ρ, (2)

1School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel.E-mail: Eugene [email protected]; [email protected]

75

0885-7474/06/0600-0075/0 © 2006 Springer Science+Business Media, Inc.

76 Kashdan and Turkel

where �J is the electric current density vector and ρ is the electric chargedensity.

For linear, homogeneous, isotropic materials we relate the magneticflux density vector �B to the magnetic field vector �H and the electric fluxdensity vector �D to the electric field vector �E using:

�B =µ �H, �D = ε �E.

We assume µ and ε are given scalar functions of space only. ε is thedielectric permittivity and µ is the magnetic permeability. Both of thesequantities are positive and describe the dielectric and magnetic character-istics of the material. In many cases ε and µ are constant within eachbody. We set ε = ε0 · εr and µ = µ0 · µr , where µ0 = 4π × 10−7 H/m andε0 = 1

c2µ0F/m are the free space permeability and permittivity respectively

(c≈3.0×108 m/sec is the speed of light). In general, the relative permittiv-ity εr and relative permeability µr are frequency dependent. However, inthis paper we concentrate on wave propagation through materials withoutsuch dependence, the so-called simple materials. The magnetic permeabilityµr is equal to one for almost all simple materials except metals which canbe considered as perfect electric conductors (PEC). The dielectric permit-tivity satisfies εr �1. It is discontinuous at the interface between materialsand these jumps frequently cause significant difficulties for numerical sim-ulations.

The ultimate goal [8] is a three dimensional time dependent codeto study electromagnetic phenomena. This necessitates the analysis of theorder of convergence, stability and robustness of the numerical schemes.To facilitate this analysis we shall consider the one-dimensional Maxwellsystem in frequency space. The results can then be applied to the timedependent code using Fourier analysis.

2. MODEL PROBLEM

The following infinite material structure (Fig. 1) can be considered amodel for the electromagnetic wave scattering by a physical body.

Where

ε(x)=

⎧⎪⎨

⎪⎩

ε1, x <L1,

ε2, L1 <x <L2,

ε1, x >L2.

We consider the physical domain [0,L]. We place material interfacescontaining regions with different relative dielectric permittivity εr at x=L1

A High-Order Accurate Method 77

Fig. 1. Model problem.

and x=L2. We assume µ=µ1=µ2 =µ0. A unit amplitude wave entersfrom −∞ travelling in the positive direction. We Fourier transform in time.This yields

iωεE − ∂H

∂x=0, (3)

−∞<x <∞,

iωµH − ∂E

∂x=0. (4)

We also consider the one-dimensional Helmholtz equation for theelectric field. It is derived by the differentiation of (4) by x and its sub-stitution into (3).

Exx + [ω2Q(x)]E =0. (5)

The multidimensional version of this is a second order linear elliptic equa-tion with variable coefficients, where

Q(x)= ε(x)µ= 1c2(x)

.

For the magnetic field, H , Maxwell’s equations can be converted intothe Helmholtz equation by the differentiation of (3) by x, yielding

1µ

∂

∂x

(1

ε(x)

∂H

∂x

)

+ω2H =0. (6)

Left of the first interface and between the two interfaces the solutionof (5) consists of two waves, travelling to the right and left. To the right


of the second interface there only exists a wave travelling in the positivedirection. We denote α(x)=√

Q(x). So,

E =

⎧⎪⎨

⎪⎩

e−iα(1)ωx +Reiα(1)ωx, x <L1,

Ae−iωα(2)x +Beiωα(2)x , L1 <x <L2,

T e−iα(1)ωx, x >L2.

(7)

At the interfaces the solution and its first derivative remain continuous.At x =L1:

{e−iωα(1)L1 +Reiωα(1)L1 =Ae−iωα(2)L1 +Beiωα(2)L1 ,

α(1)(−e−iωα(1)L1 +Reiωα(1)L1)=α(2)(−Ae−iωα(2)L1 +Beiωα(2)L1).(8)

At x =L2:

{Ae−iωα(2)L2 +Beiωα(2)L2 =T e−iωα(1)L2 ,

α(2)(−Ae−iωα(2)L2 +Beiωα(2)L2)=−α(1)T e−iωα(1)L2 .(9)

This gives a system of four equations with four unknowns. Solving it weget:

T =4α(1)

α(2) e−iωα(1)(L1−L2)

D,

A =2α(1)

α(2)

(1+ α(1)

α(2)

)e−iω(α(1)L1−α(2)L2)

D,

B =2α(1)

α(2)

(1− α(1)

α(2)

)e−iω(α(1)L1+α(2)L2)

D,

R =

(

1−(

α(1)

α(2)

)2)

e−2iωα(1)L1

{eiωα(2)(L1−L2) − e−iωα(2)(L1−L2)

}

D,

where

D =(

1+ α(1)

α(2)

)2

e−iωα(2)(L1−L2) −(

1− α(1)

α(2)

)2

eiωα(2)(L1−L2)

One can similarly derive the exact solution for the magnetic field Eq. (6).


3. REGULARIZATION OF DISCONTINUOUS PERMITTIVITY ε

3.1. Background

There are several ways to treat a discontinuity. The important ques-tion is how to preserve the global order of accuracy for locally high-orderaccurate schemes. One of the approaches to the solution of Maxwell equa-tions with discontinuous coefficients is based on one-sided finite differenceformulae, approximating the differential equation from both sides of theinterface (see e.g. [3]). However, for a multidimensional problem it is diffi-cult to achieve higher order accuracy for an interface not aligned with thegrid. Another drawback of this approach is the violation of the Gauss’ lawat the interface that can lead to the spurious solutions (see e.g. [7]).

An alternative approach is “regularization” (see e.g. [10]). We replacethe discontinuous functions, ε, µ by a continuous approximation. Wethen use the standard scheme everywhere. It is straightforward to verifythat this automatically preserves a zero divergence (in multidimensionalcase) when the system is solved by a central difference scheme even forspace dependent coefficients. We shall develop our algorithm based on thisapproach.

One can regularize either ε and µ or else 1/ε and 1/µ. Regularizationcan be viewed as a generalized averaging of the piecewise-continuous func-tion. Algebraic, geometric and harmonic averages have different effects onthe numerical solution. Wesseling [12] has shown that arithmetic averagingof the piecewise-continuous coefficient a(x) in equation

d

dx

[

a(x)d

dx

]

u(x)=f

reduces the second order of a numerical scheme to the first order. In con-trast, the geometric average preserves the second order accuracy of thescheme independent of the location of the discontinuity relative to the grid.

For multidimensional problems we need to consider the angle of inci-dence of the electromagnetic wave to the interface. In [1] it is shown for thetwo-dimensional Maxwell equations that the Yee scheme preserves the secondorder of accuracy if the angle of incidence is less than 90◦ and the harmonicaverage is used for the approximation of the discontinuity in the dielectric per-mittivity ε. If the angle of incidence is more than 90◦, the arithmetic averageshould be used. It is known that electric and magnetic field vectors are perpen-dicular each other. So one should use different types of averaging when both ε

and µ are discontinuous at the interface. We shall only consider a discontinuityin ε and assume the electric field is one dimensional.

Once regularization occurs within a mesh width, it is equivalent tothe solving the discontinuous problem. We compare different techniques


for the approximation of the piecewise-continuous dielectric permittivity.We divide these techniques into local and global methods. We pay atten-tion to monotonicity.

3.2. Local Regularization

We regularize ε1 and ε2 near the discontinuities. We choose two pointsL1 − δ and L1 + δ on each side of the first interface and L2 − δ and L2 + δ

on each side of the second interface. We choose δ small enough so thatL1 + δ <L2 − δ so that the two interfaces do not overlap. We connect ε onthe two sides of the discontinuity with smooth functions. We locate the inter-face at a node and connect the ends with a Hermite cubic spline. A cubicspline can be written as

S3(x, δ)= c1 + c2(x −xk)+ c3(x −xk)2 + c4(x −xk)

3. (10)

We calculate in the interval xk−1/2 �x �xk+1/2 the following parameters:

c1 = fk+1/2,

c2 = f ′k+1/2,

c3 =3Sk −f ′

k+1/2 −2f ′k−1/2

∆x,

c4 = −2Sk −f ′

k+1/2 −f ′k−1/2

(∆x)2

and

Sk = fk+1/2 −fk−1/2

∆x,

where f is equal to ε or 1/ε at the half-nodes. Hyman [6], shows that thisis the most accurate approximation that can be achieved when the deriva-tive, f ′, is approximated using a high-order accurate implicit finite differencescheme.

One can try to improve the quality and accuracy of the approxima-tion by demanding monotonicity. The following criteria, [6], allows theaddition a monotonicity restraint (MR) to the approximation

f ′k =

⎧⎪⎨

⎪⎩

min[max(0, f ′k),3 min(Sk−1, Sk)], min(Sk−1, Sk)>0,

max[min(0, f ′k),3 max(Sk−1, Sk)], max(Sk−1, Sk)<0.

0, Sk−1 ·Sk �0(11)

As an alternative to a cubic spline one can consider a high-orderpolynomial smoothly connecting the known εl and εr at a distance δ from


the interfaces. Let z=x −x0/δ be the normalized distance from the inter-face x0. We define the polynomial

f (x)= fl +fr

2+ fl −fr

2315128

z

(

1− 4z2

3+ 6z4

5− 4z6

7+ z8

9

)

(12)

f (x) and its first four derivatives are continuous at x0 − δ and x0 + δ. More-over, this function is monotone since f ′ = 315

128 (1− z2)4 �0 for −1� z�1.

3.3. Global Regularization

For global regularization (2δ =L) we choose an implicit fourth-orderaccurate (Pade) interpolation. Based on the Taylor expansion we can write:

fi+1/2 +fi−1/2

2=fi + h2

8 f′′i +O(h4),

fi+1 +fi−1

2=fi + h2

2 f′′i +O(h4).

Eliminating the error term multiplied by h2 we obtain

fi+1 +fi−1

8+ 3

4fi =

fi+1/2 +fi−1/2

2+O(h4). (13)

Equation (13) is shown in matrix form in Fig. 2. f is equal to eitherε or 1/ε. This regularization does not necessarily preserve monotonicity. Ituses all the grid points to construct a smooth approximation of the piece-wise-continuous function. Similarly the values of ε are changed everywhereeven far from the discontinuity.

One can construct a monotonic global regularization using sums ofhyperbolic tangent functions. We approximate ε(x) by

ε(x, η)= c+dtanh[η(x −L1)]− tanh[η(x −L1)]

tanh(ηL1)+ tanh(ηL2), −∞<x <∞, (14)

Fig. 2. Implicit interpolation.


0 0.5 1 1.5 2 2.5 31

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

x

ε (x

, η)

η = 1η = 10η = 100

Fig. 3. Approximation of ε by continuous function in model problem.

where η is large and c and d are scaling parameters. Matching at ±∞ weget c = ε1 and d = ε2 − ε1. In Fig. 3 we present ε as a function of x forvarious η.

4. MATCHING CONDITION

We write (5) in operator form as

Lu=0, (15)

where there is a discontinuity in the coefficients of L. We replace L by itsregularized approximation Lδ, where 2δ is the length of the zone wherethe discontinuous function is replaced by its regularization. The regular-ized equation is given by

Lδuδ =0. (16)


The regularization is called global if 2δ is equal to the length of thephysical domain and local otherwise. To solve (15) numerically we approx-imate (16) by a discretization yielding

Lδ(h)uδ(h) =0. (17)

The regularization parameter δ may depend on the mesh size h.The numerical implementation introduces two types of errors: E1 =

‖u − uδ‖ – the error caused by replacement of the discontinuous prob-lem by the regularized problem (“analytic error”) and E2 =‖uδ −uδ(h)‖ –the error from the numerical discretization of the regularized problem(“numerical error”). The total error is bounded by

‖u−uδ(h)‖�‖u−uδ‖+‖uδ −uδ(h)‖=E1 +E2. (18)

Clearly, E1 becomes smaller when the length of the regularizationregion decreases (δ → 0). However, δ → 0 increases the error E2. We con-sider a part of the model problem that includes the first interface onlylocated, for simplicity, at x =0 (Fig. 4).

For δ 1 ε near the interface can be approximated by a linear func-tion.

ε(x)=

⎧⎪⎨

⎪⎩

ε1, x <−δ,ε2−ε1

2δx + ε2+ε1

2 , −δ <x <δ,

ε2, x >δ.

Fig. 4. Model problem with one interface.


Then Eq. (5) becomes⎧⎪⎨

⎪⎩

Exx +ω2Q1E =0, x <−δ,

Exx +ω2Q(x)E =0, −δ <x <δ,

Exx +ω2Q2E =0, x >δ.

(19)

The second equation of (19) can be transformed into the Airy equation.Define new variables: k1=ω2ε1µ and k2=ω2ε2µ. Then, the solution of (19)is given by

E(x)=

⎧⎪⎨

⎪⎩

e−i√

k1x +Rδei√

k1x, x <−δ,

A ·Ai(ξ(x))+B ·Bi(ξ(x)), −δ <x <δ,

Tδe−i

√k2x, x >δ,

where Rδ and Tδ are the reflection and transmission coefficients. Ai andBi are the Airy functions of the first and second kind respectively and

ξ(x) = −(

2δ

k2 −k1

) 23(

k2 −k1

2δx + k1 +k2

2

)

So ξ(x)→0 as δ →0 for −δ <x <δ.We match the solution and its derivative at x =−δ and x = δ.

ei√

k1δ +Rδe−i

√k1δ = AAi(ξ(−δ))+B Bi(ξ(−δ)),

−i√

k1

[ei

√k1δ −Rδe

−i√

k1δ]

= −(

k2 −k1

2δ

)13 [

AAi′(ξ(−δ))+B Bi′(ξ(−δ))],

(20)

Tδe−i

√k2δ = AAi(ξ(δ))+B Bi(ξ(δ)),

−i√

k2Tδe−i

√k2δ = −

(k2 −k1

2δ

) 13 [

AAi′(ξ(δ))+B Bi′(ξ(δ))],

where

ξ(−δ)=−(

2δ

k2 −k1

) 23

k1, ξ(δ)=−(

2δ

k2 −k1

) 23

k2. (21)

This yields four equations for Rδ, Tδ, A and B. The solution is given inthe Appendix A.

From the continuity of the solution across the interface follows thatreflection and transmission coefficients are connected by the matching con-dition T −R = 1. Based on the solution of (20), we calculate Tδ −Rδ andexpand it into the Taylor series in δ. The first terms of the expansion are


Tδ −Rδ =1− (√

k2 −√k1)

√k1

3δ2 +O(δ4). (22)

We see that the matching condition is satisfied, when δ → 0. The errordecays as O(δ2). Since, on the discrete level δ = δ(h), this limits the accu-racy as a function of h. Hence, we do not expect an asymptotic accuracybetter than O(h2). Specifically, for coarse meshes the regularization erroris small compared with the truncation error. Thus, for engineering accu-racy the fourth order implicit method is much more efficient than the Yeescheme. However, for finer grids the regularization error dominates sincethe optimal δ is less than one mesh width wide. Hence, asymptotically weexpect at most a second order scheme.

5. NUMERICAL ALGORITHM

5.1. Artificial Boundary Conditions

We consider the model problem with a boundary condition, which isan incoming wave at −∞. We introduce the scattered field Escat (x) suchthat

E(x)=Escat (x)+ e−iωα(1)x .

After substitution into (5) Escat satisfies:

Escatxx +ω2Q(x)Escat =g(x), (23)

where g(x) = ω2[(

α(1))2 −Q(x)

]e−iωα(1)x . Outside the physical domain

Escat satisfies the Sommerfeld boundary conditions: there are no wavesreturning from ±∞ into the physical domain.

The system of Maxwell equations ((3), (4)) has the same analyticsolution as the Helmholtz equation derived for E and for H . However,it differs on the numerical level. The scattered fields satisfy an inhomoge-neous linear system of ODEs

iωε(x)Escat − ∂Hscat

∂x=f (x),

−∞<x <∞, (24)

iωµHscat − ∂Escat

∂x=0,

where f (x)= iω[ε1 − ε(x)]e−iωα(1)x .


Fig. 5. Algorithm for solution of frequency dependent Maxwell equations.

To solve the system (24) numerically we need to bound the domainand construct boundary conditions at the artificial surface. We use the uni-axial PML as described in [4]. Inside the physical domain we solve theoriginal system (24) and in the PML we solve a transformed system:

iωεSEscat − ∂Hscat

∂x=0,

−∞<x <∞, (25)

iωµSHscat − ∂Escat

∂x=0,

where S =1 + σiω

. σ = σε

ε= σµ

µis a polynomial inside the PML and zero

inside the physical domain. At the interface between the PML and theinterior σ must be sufficiently smooth.

For the one-dimensional problem one may use a characteristic orlocal [2] boundary condition. However, since we wish to model a multi-dimensional problem with a high-order scheme then the use of the PMLis preferable [9] (Fig. 5).

5.2. Finite Difference Discretization

The scheme we use to approximate derivatives is a staggered compactimplicit scheme. Gottlieb and Yang [5] and Turkel [11] have shown that astaggered scheme is more accurate and efficient than a co-located schemefor the same order of accuracy. The electric field component E is locatedat the nodes and the magnetic field component H is located at half-nodes.We place all interfaces at a node. To establish notation, we write the


second-order accurate spatial derivative operator as (Du)i . (Du)i ≈ ∂u(xi )∂x

with a local truncation error of order O(∆x2).The scheme is derived from Ty operator [11]:

(Du)i+1 + (Du)i−1

24+ 11

12(Du)i =

ui+ 1

2−u

i− 12

∆x. (26)

We denote a Ty operator applied to the approximation of the spatialderivatives at nodes as AE , and at half-nodes as AH . It yields

∂Hscat

∂x= (AH )−1D(Hscat ),

∂Escat

∂x= (AE)−1D(Escat ).

Based on the notation introduced in [8], we define S = iωε(x)S and S =iωµS. After discretization, the system (24)+(25) becomes

AH · (SEscat )−D(Hscat ) = AH ·f,

AE · (SH scat )−D(Escat ) = 0.

This scheme is locally fourth order accurate both inside the physicaldomain and inside the PML. Using the fact that E is located at the nodes andH is at the half-nodes we construct a pentadiagonal system of linear equations:

6. NUMERICAL EXPERIMENTS

6.1. Setup

For the model problem there is an explicit solution of (3,4) which fol-lows from the solution of the Helmholtz equation. Let L=3 m, L1 =0.75 mand L2 =2.25 m. We choose εr =2 between the interfaces. εr =1 to the leftof the first interface and to the right of the second interface. The wave-length λ in free space is 0.3 m and ω=2π GHz. Between the interfaces thewavelength is reduced by a factor

√2. The electric component of the solu-

tion is shown in Fig. 6.

6.2. Global Regularization

For global regularization (13) the errors, in the L2 norm, are pre-sented in Table I.

Computing on even finer meshes confirms that asymptotically therate of accuracy approaches 2. However, this occurs for error levels much


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3−4

−2

0

2

4

Fig. 6. Amplitude (top) and argument (bottom) of solution to model problem.

Table I. Error of Global (Implicit) Regularization

No. of nodes ∆x ‖Escat −Escatδ(h)‖L2 Rate Order of accuracy

129 0.0234 1/ε: 2.0058×10−2

ε: 2.0484×10−2

257 0.0117 1/ε: 5.2048×10−3 3.8538 1.9463ε: 1.9044×10−3 10.7561 3.4271

513 0.0059 1/ε: 3.6608×10−3 1.4218 0.5077ε: 3.3058×10−4 5.7608 2.5263

smaller than usual accuracy levels. For practical error levels we get moreaccurate results with regularization than with arithmetic averaging acrossthe interface (see [1, 8]). Numerical computations [8] show that using (14)yields a larger error than the (Pade) approximation (5). Regularizationbased on 1/ε yields a much larger error than regularization based on ε.

In Fig. 7 we show the relative (pointwise) error for a grid with 257nodes for the regularization of both ε and 1/ε. The largest relative erroroccurs near the first interface. The regularization of 1/ε yields a much


0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04R

elat

ive

erro

r

x

1/εε

Fig. 7. Relative pointwise error of solution using global regularization.

larger error at all grid points. One observes that the major errors are inthe vacuum region.

6.3. Local Regularization

We choose a Hermite cubic spline as the connecting function for thelocal regularization of ε and 1/ε. In Table II we present the total error forvarious grids. We fix the length of the δ-interval and check the influenceon the accuracy of the MR to the spline approximation. Though the reg-ularized variable is close to constant on either side of the discontinuity itis now smooth. The results are now insensitive to the value of δ.

We see from Tables I and II that a global regularization yields betterresults than a local regularization. We also observe that regularization ofε yields more accurate results than the regularization of 1/ε. The additionof a MR to the regularization does not make any significant changes tothe accuracy.

It is shown in [8] that in using of regularization (14) the best choicefor η is sufficiently large. So the gradient of the hyperbolic tangent func-tion lies within the closest grid point to the discontinuity. Hence, this


Table II. Error of the Local Regularization

δ (cm) No. of nodes ∆x ‖Escat −Escatδ(h)‖L2 Rate Order of accuracy

129 0.0234 1/ε: 5.3177×10−2

ε: 2.0667×10−2

9.375 257 0.0117 1/ε: 3.2576×10−2 1.6323 0.7070ε: 2.6548×10−3 7.7848 2.9607

513 0.0059 1/ε: 1.6468×10−2 1.9781 0.9841ε: 6.4382×10−4 4.1235 2.0439

129 0.0234 1/ε: 5.4096×10−2

ε: 2.0821×10−2

9.375 257 0.0117 1/ε: 3.2833×10−2 1.6476 0.7204

+MR ε: 2.7040×10−3 7.7001 2.9449513 0.0059 1/ε: 1.6530×10−2 1.9863 0.9841

ε: 6.4904×10−4 4.1335 2.0473

simply duplicates arithmetic averaging of ε at the discontinuity. Similarlyfor the high-order polynomial (12) the best choice is δ =1, i.e. we changeonly the discontinuity point itself. However, as shown in [8] the simplearithmetic averaging yields less accurate results than both global (implicit)and local (Hermite spline based) regularizations.

7. ERROR ANALYSIS

7.1. Analytic Error

There are two main sources of error. First is the replacement of the con-tinuous problem by a discretization. The second is the replacement of thepiecewise smooth function ε by a regularized smooth function. Let Escat bethe exact solution to the continuous problem. Define E∗ be the exact solu-tion to the regularized differential equation and Escat

δ the solution of theregularized difference equation. We note that in general the optimal δ is afunction of the mesh width h. The first error is given by E1 =‖Escat −E∗‖.The second error is given by E2 =‖E∗ −Escat

δ ‖. We investigate the role ofthe analytic error E1 as a part of the total error E. We need an “exact”solution to the regularized differential equation which has variable coeffi-cients. We get this by computing on a fine mesh. We construct a “referencesolution” E∗ for the electric field component in (24) using a grid of 2049nodes. We need to choose the PML thick enough so that it does not affectthe accuracy. For grids coarser than 2049 nodes we use a PML with 8 meshpoints. For the finest grid of 2049 we used a PML with 16 mesh points.


Table III. Numerical Error of Global Regularization vs. Total Error

Order OrderNo. of nodes ∆x ‖Escat −Escat

δ ‖L2 of accuracy ‖E∗ −Escatδ ‖L2 of accuracy

129 0.0234 2.0484×10−2 2.0544×10−2

257 0.0117 1.9044×10−3 3.4270 1.8800×10−3 3.4499513 0.0059 3.3058×10−4 2.5263 3.1236×10−4 2.5895

Table IV. Numerical Error of Local Regularization vs. Total Error

Order OrderNo. of nodes ∆x ‖Escat −Escat

δ ‖L2 of accuracy ‖E∗ −Escatδ ‖L2 of accuracy

129 0.0234 2.0667×10−2 2.0667×10−2

257 0.0117 2.6548×10−3 2.9606 2.6101×10−3 2.9852513 0.0059 6.4382×10−3 2.0439 6.0488×10−4 2.1094

Therefore, the inequality (18) can be rewritten as

E =‖Escat −Escatδ ‖�‖Escat −E∗‖+‖E∗ −Escat

δ ‖=E1 +E2.

We only regularize ε. In the Tables III and IV we compare the total errorfrom Table I for the global and local regularization with the estimatednumerical error.

For coarse grids the error of the numerical solution of the regular-ized problem is the main component of the total error. The error causedby the regularization plays only a minor role in the total error. For finegrids both E1 and E2 are the same order of magnitude and so the totalerror decays slower than the formal order of accuracy of the scheme.

7.2. Error in Different Regions

We now study the accumulation of errors at the interface. We com-pare the error outside the interfaces vs. the error between the interfacesboth local and global regularizations of ε. We consider the global regu-larization and the local regularization inside the interval δ =9.375 cm.

In Tables V and VI we compare the error outside the interfaces withthe error between the interfaces. Between the interfaces the order of accu-racy is about third order for the global interpolation and about 2 1

2 for the


Table V. Error in Different Regions for Global Regularization

Order OrderNo. of nodes ∆x ‖Outside‖ of accuracy ‖Between‖ of accuracy

129 0.0234 2.3128×10−2 1.7391×10−2

257 0.0117 1.9261×10−3 3.5859 1.8822×10−3 3.2078513 0.0059 3.9289×10−4 2.2935 2.5305×10−4 2.8995

Table VI. Error in Different Regions for Local Regularization

Order OrderNo. of nodes ∆x ‖Outside‖ of accuracy ‖Between‖ of accuracy

129 0.0234 2.0970×10−2 2.0354×10−2

257 0.0117 2.7011×10−3 2.9567 2.5810×10−3 2.9647513 0.0059 7.9532×10−4 1.7639 4.4228×10−4 2.5595

local regularization. Outside the interfaces the order of accuracy decreasesbecause of the PML.

8. CONCLUSIONS

We have performed detailed computations for the Maxwell system inone dimension in frequency space. The equations are solved with a fourthorder accurate compact implicit method on a staggered grid. ε is discon-tinuous across interfaces but is replaced by a continuous function. A globalregularization of ε based on a Pade approximation yields a lower total errorcompared with local regularization based Hermite splines. Nevertheless, forvery fine grids the order of accuracy reduces to second order. At engineer-ing accuracy both methods combined with the fourth order scheme are sig-nificantly more accurate than simple averaging of ε with the standard Yeescheme. This is analyzed by considering a simple local regularization. Anexplicit solution shows that δ = O(h2) and so for fine enough meshes theoptimal width for a local regularization is less than one mesh width. Thiscan not be represented with a standard finite difference scheme.

For coarse and intermediate grids the numerical error dominates theregularization error. Furthermore, the error in the interior behaves betterthan the error outside the interfaces. This indicates that the PML contrib-utes to the error and reduces the total accuracy.


APPENDIX A. COMPUTATION OF THE MATCHING CONDITION

Solving system (20) with Mathematica we get

Rδ =2e2iδ(P1 +P2), (A.1)

where

P1 ={

(k2 −k1)Ai′(ξ(−δ))(Ai

′(ξ(−δ))Bi

′(ξ(δ))−Bi

′(ξ(−δ))Ai

′(ξ(δ)))+2iδ

×(Ai(ξ(−δ))k1√

k2(Bi(ξ(−δ))Ai(ξ(δ))−Ai(ξ(δ))Bi(ξ(−δ)))

}/

Denominator

P2 = 2δ

(k2 −k1

2δ

) 13{

Ai(ξ(−δ))k1(Ai(ξ(−δ))Bi′(ξ(δ))−Bi(ξ(−δ))Ai

′(ξ(δ)))

+Ai(ξ(δ))√

k1k2(Bi(ξ(−δ))Ai′(ξ(−δ))−Ai(ξ(−δ))Bi

′(ξ(−δ)))i

(k2−k1

2δ

) 13

+[√

k1Ai′(ξ(δ))(Bi(ξ(−δ))Ai

′(ξ(−δ))−Ai(ξ(−δ))Bi

′(ξ(−δ)))

√k2Ai

′

+(ξ(−δ))(Bi′(ξ(−δ))Ai(ξ(δ))−Ai

′(ξ(−δ))Bi(ξ(δ)))

]}/

Denominator

and

Tδ ={

−8δeiδ(√

k1+√

k2)

(k2 −k1

2δ

) 13

(Bi(ξ(δ))Ai′(ξ(δ))−Ai(ξ(δ))Bi

′(ξ(δ)))

×Ai(ξ(−δ))k1 + i

(k2 −k1

2δ

) 13

Ai′(ξ(−δ))

√k1

}/

Denominator. (A.2)

The Denominator is given by

Denominator = 2(k2 −k1)Ai′(ξ(−δ))(Ai

′(ξ(−δ))Bi

′(ξ(δ))−Bi

′(ξ(−δ))Ai

′(ξ(δ)))

+4iδ(Ai(ξ(−δ))k1

√k2(Bi(ξ(−δ))Ai(ξ(δ))−Ai(ξ(δ))Bi(ξ(−δ)))

+4δ

(k2−k1

2δ

)13{

Ai(ξ(−δ))k1(Ai(ξ(−δ))Bi′(ξ(δ))−Bi(ξ(−δ))Ai

′(ξ(δ)))

+√

k1k2(Bi(ξ(−δ))Ai′(ξ(−δ))Ai(ξ(δ))+Ai(ξ(−δ))Bi

′(ξ(−δ))Ai(ξ(δ))

−2Ai(ξ(−δ))Ai′(ξ(−δ))Bi(ξ(δ)))+ i

(k2 −k1

2δ

) 13

×[√

k1(Bi(ξ(−δ))Ai′(ξ(−δ))Ai

′(ξ(δ))


+Ai(ξ(−δ))Bi′(ξ(−δ))Ai

′(ξ(δ))−2Ai(ξ(−δ))Ai

′(ξ(−δ))Bi

′(ξ(δ)))

+√

k2Ai′(ξ(−δ))(Ai

′(ξ(−δ))Bi(ξ(δ))−Bi

′(ξ(−δ))Ai(ξ(δ)))

]}

.

Expanding the Airy functions and their derivatives in a Taylor series in δ

we get

Ai(ξ(−δ)) = 1

323 Γ( 2

3

) + 223 k1

313 Γ( 1

3

)

(1

k2 −k1

) 23

δ23 − 2k3

1

353 Γ( 2

3

)

(1

k2 −k1

)2

δ2

− 223 k4

1

343 Γ( 1

3

)

(1

k2 −k1

) 83

δ83 +O(δ4),

Bi(ξ(−δ)) = 1

316 Γ( 2

3

) − 223 3

16 k1

Γ( 1

3

)

(1

k2 −k1

) 23

δ23 − 2k3

1

376 Γ( 2

3

)

(1

k2 −k1

)2

δ2

+ 223 k4

1

356 Γ( 1

3

)

(1

k2 −k1

) 83

δ83 +O(δ4),

Ai′(ξ(−δ)) = − 1

313 Γ( 1

3

) + 213 k2

1

323 Γ( 2

3

)

(1

k2 −k1

) 43

δ43 + 4k3

1

343 Γ( 1

3

)

(1

k2 −k1

)2

δ2

− 273 k5

1

5 ·353 Γ( 2

3

)

(1

k2 −k1

) 103

δ103 +O(δ4),

Bi′(ξ(−δ)) = 3

16

Γ( 1

3

) + 213 k2

1

316 Γ( 2

3

)

(1

k2 −k1

) 43

δ43 − 4k3

1

356 Γ( 1

3

)

(1

k2 −k1

)2

δ2

− 273 k5

1

5 ·376 Γ( 2

3

)

(1

k2 −k1

) 103

δ103 +O(δ4).

Expansions for Ai(ξ(δ)), Bi(ξ(δ)), Ai′(ξ(δ)) and Bi

′(ξ(δ)) can be con-

structed similarly, with the replacement of k1 by k2 according to (21).Using the expansions (A.1, A.2) we get approximations for Rδ and Tδ. Let

P3 = 2i

15(√

k1 +√k2)

4

(

−k321 k2

2 +k121 k3

2 +2k31k

122 +k

721 +2k1k

522 −4k2

1k322 −k1

52 k2

)

Then

Rδ =√

k1 −√k2√

k1 +√k2

− 2

3(√

k1 +√k2)

3

(

k21k

122 −k

121 k2

2 +k21k

122 −k

121 k2

2

)

δ2

−P3δ3 +O(δ4)=R +O(δ2)


and

Tδ = 2√

k1√k1 +√

k2− 1

3(√

k1 +√k2)

3(2k

321 k2 −k

521 −k

121 k2

2)δ2

−P3δ3 +O(δ4)=T +O(δ2).

REFERENCES

1. Andersson, U. (2001). Time-Domain Methods for the Maxwell’s Equations. Ph.D. thesis,Royal Institute of Technology – Sweden.

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A High-Order Accurate Method for Frequency Domain Maxwell Equations with Discontinuous Coefficients

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