Accurate implementation of current sourc

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Accurate implementation of current sources in
the ADI-FDTD Scheme
Article in IEEE Antennas and Wireless Propagation Letters · January 2005
DOI: 10.1109/LAWP.2004.831078 · Source: IEEE Xplore

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 3, 2004

141

Accurate Implementation of Current Sources in the

ADI–FDTD Scheme
S. González García, Member, IEEE, A. Rubio Bretones, Senior Member, IEEE,
R. Gómez Martín, Senior Member, IEEE, and Susan C. Hagness, Member, IEEE

Abstract—This letter presents a new method of implementing
current sources in the alternating-direction-implicit finite-difference time-domain (ADI–FDTD) method. The current source
condition is derived systematically, starting with the fully implicit
Crank–Nicolson (CN) FDTD scheme for Maxwell’s curl equations
and building the ADI-FDTD scheme as a second-order-in-time
perturbation of the CN-FDTD scheme. We demonstrate that the
accuracy of this new source condition is superior to previously
proposed techniques.
Index Terms—Alternating-direction implicit (ADI) methods,
finite-difference time-domain (FDTD) methods, incident wave
source conditions, numerical stability.

I. INTRODUCTION
HE
alternating-direction-implicit
finite-difference

time-domain (ADI–FDTD) method is an unconditionally stable alternative to the standard fully explicit FDTD
method [1], [2]. Although its range of applicability, especially
to low-frequency problems, needs further development [3],
ADI–FDTD has been successfully applied to several problems of interest where very fine meshes with respect to the
wavelength are required for at least part of the computational
domain. The ADI–FDTD scheme involves two sub-iterations,
to
the first of which advances the fields from time step
, and the second of which advances the fields from
to
. The magnetic (or electric) field updating
expressions remain explicit while the electric (or magnetic)
fields are calculated using implicit updating equations along
directions through the grid that alternate from one subiteration
to the next. Since the implicit equations represent a tridiagonal
matrix system, the ADI–FDTD method offers unconditional
numerical stability with modest computational overhead.
The standard FDTD scheme permits the use of explicit wave
source conditions wherein specific electric and/or magnetic
field components at source points in the grid are updated separately from the point-by-point explicit Yee updating equations.

Several papers have proposed similar source conditions for

T

Manuscript received January 27, 2004; revised April 14, 2004. This work was
supported by the Spanish National Research Project TIC-2001-3236-C02-01
and TIC-2001-2364-C03-03, and by the National Science Foundation Presidential Early Career Award for Scientists and Engineers (PECASE) ECS-9985004.
S. G. García, A. R. Bretones, and R. G. Martín are with the Departimento
Electromagnetismo y Física de la Materia, Facultad de Ciencias, University of
Granada, 18071 Granada, Spain (e-mail: salva@ugr.es).
S. C. Hagness is with the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706 USA (e-mail:
hagness@engr.wisc.edu).
Digital Object Identifier 10.1109/LAWP.2004.831078

ADI-FDTD [1], [4], [5]. However, in [6], it was pointed out that
these explicit types of source conditions in ADI-FDTD lead
to localized errors. If, for example, the ADI-FDTD algorithm
is formulated so that the electric fields are updated implicitly,
then the electric current source excitation function should be
embedded in the known column vector on the right-hand side

of the tridiagonal matrix system for the -, -, or -directed
lines that pass through the location of the current source.
A remaining issue regarding the implementation of current
sources in the ADI-FDTD scheme is the lack of consensus in
the literature about the discrete-time values that should be used
to evaluate the current source excitation function in each of
the two subiterations of the ADI-FDTD scheme. In this paper,
we address this issue and present for the first time a rigorously
derived formulation for current sources in the ADI–FDTD
method. In Section II, we consider the ADI-FDTD algorithm as
a second-order-in-time perturbation of a Crank–Nicolson (CN)
FDTD scheme for Maxwell’s curl equations. This systematic
framework permits the introduction of current source terms in
a straightforward manner. In Section III, we briefly consider
the consistency of the proposed method for implementing
current sources and compare its accuracy to previously published approximations. Finally, in Section IV, we present some
illustrative results of numerical experiments.
II. ADI–FDTD WITH CURRENT SOURCES
The time-dependent Maxwell’s curl equations for linear,
isotropic, nondispersive, lossy media can be written in Cartesian coordinates as

(1)
where

is the composite electromagnetic field vector
(2)

and is a source vector comprised of free electric and magnetic
current densities
(3)
with the superscript
mensional operator,

1536-1225/04$20.00 © 2004 IEEE

denoting a matrix transpose. The six–di, can be written in block matrix form as
(4)

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 3, 2004

where , , , and
are the permittivity, permeability, electric
is
conductivity, and equivalent magnetic loss, respectively,
the 3 3 identity matrix, and is the curl operator
(5)
represents the partial derivative with respect to
The operator
.
First, we consider a fully implicit CN-FDTD solution to (1)
because of its close relationship to the ADI-FDTD scheme. The
CN-FDTD scheme is obtained by replacing the derivatives with
second-order-accurate centered finite differences on a standard
staggered spatial Yee lattice [7] with all fields co-located in time.
in time; thereThe finite differences are centered at
fore, the field vector on the right-hand side of (1) is averaged
in time for consistency. The resulting finite-difference equation
can be expressed as
(6)

where noncalligraphic symbols have been used to represent all
numerical variables and operators. For example,
represents
represents the finitethe numerical version of . Similarly,
difference version of
.
The CN–FDTD scheme is unconditionally stable regardless
of the ratio between the space and time increment employed [8].
However, its application to practical problems requires prohibitively large computational resources because it requires the solution of a sparse linear system of equations at each time step.
Using the approach of [9], [10] originally employed for parabolic equations, we previously demonstrated that it is possible
to factorize the CN–FDTD scheme into a two-step procedure
, with each
for updating the fields from time step to
subiteration requiring the solution of a tridiagonal system of
equations [3]. This factorization is made possible by adding a
second-order perturbation term to (6) in the following manner:

has been introduced as the auxiliary intermediate
where
vector field and represents the 6 6 identity operator. Note

in the first subthat the fields are updated from to
iteration, and from
to
in the second subiteration.
The two-step ADI-FDTD scheme is obtained from (9) and
(10) using the following definitions for and :
(11)
and
are the numerical versions of the even and odd
where
parts of the analytical curl operator,

(12)
Assuming the case of lossless media for simplicity, the resulting
electric and magnetic field update equations are
(13)
(14)

(15)

(16)
This algorithm can be simplified by substituting (14) into (13)
and (16) into (15), yielding an implicit (tridiagonal) updating
scheme for the electric field and an explicit updating scheme
for the magnetic field at each subiteration.

(17)
(7)
(18)
If the operators and
(7) can be rewritten as

are chosen so that

, then

(8)

(19)

and factorized into a two-step procedure as follows:
(9)

(10)

(20)
Note that without the current source terms, (17)–(20) reduce to
standard source-free ADI-FDTD equations reported in [11].

GARCÍA et al.: ACCURATE IMPLEMENTATION OF CURRENT SOURCES IN THE ADI–FDTD SCHEME

143

III. CONSISTENCY AND ACCURACY
Our proposed method for the temporal sampling of the currents sources is evident in (9) and (10). The first subiteration (9)
approximates the time derivatives and the fields at
while the currents are taken at
.
The second sub-iteration (10) approximates the time derivatives
keeping the currents at
and the fields at
. This suggests that the update equations in each
subiteration are not consistent with Maxwell’s curl equations;
that is, they do not approximate the curl equations when all of
the space and time increments tend to zero. However, the overall
scheme (7) is still consistent with (1) up to second order both in
time and in space. This can be shown by obtaining the truncation error in the same manner used for the source–free lossless
case in [3].1 It is interesting then to note that each subiteration
in the time-stepping algorithm can lose consistency, without impacting the overall consistency of the total scheme (see [13] for
further discussion).
An alternative method for the temporal sampling of the current sources is found in [6]. In this alternative formulation, the
in the first subiteracurrents are evaluated at
in the second subiteration, thereby
tion and at
maintaining consistency within each subiteration:

(21)

(22)
between (21) and (22) yields the following
Eliminating
equivalent single-step scheme:

(23)
A comparison of (23) and (7) shows that the underbraced term in
term in (7). In fact, the
(23) is an approximation to the
underbraced term approaches
in the limit as the time
increment tends to zero. Consequently, although each subiteration in this alternative approach is consistent with Maxwell’s
curl equations, the currents are implemented in a less accurate
manner than in the new formulation proposed in the Section II.

Fig. 1. Illustration of the concept of the Huygens’ box, shown in the xy -plane
of a three-dimensional ADI-FDTD grid.

present are those requiring a separation between a total and
a scattered field zone. For these problems, the equivalence
principle is used to define on a closed Huygen’s surface a set
of equivalent currents
and
so that the interior/exterior
original sources of the problem can be removed. These currents
are so defined to create the same fields as the original sources
outside/inside, and null fields inside/outside (respectively). In
order to implement the currents on the Huygen’s surface, the
and
components are interleaved in the Yee grid in a similar
and components, which results in two
way to that of the
Huygen’s surfaces separated by a distance of half a cell. As
,
,
an example, Fig. 1 shows the positions of , ,
and
at the
–plane. This method of implementing
the equivalence principle in an FDTD computer code was
first introduced in [12]; it does not require an averaging of the
magnitudes involved.
In the first numerical experiment, we have placed on the surface of an empty Huygens’ box the currents necessary to create
propaa 10-GHz plane wave of amplitude
gating inside the box along the –axis, and a null field outside.
The cubic Huygen’s box spans 20 grid cells in each direction.
A grid resolution of 60 cells/wavelength (
) is chosen for this experiment. We define the average volumetric electric energy density at a point as follows:

(24)
IV. NUMERICAL RESULTS
To illustrate the accuracy of the current source condition
proposed in Section II, we present the results of two numerical
experiments that include both electric and magnetic current
sources. Typical problems where both types of currents are
1An extended discussion on the truncation error topic is found there, including
an analysis of the accuracy limitations of ADI-FDTD due to presence of error
terms that depend on the square of the time increment multiplied by the spatial
derivatives of the field.

Fig. 2 shows the square root of the average energy density
at a point outside the Huygens’ box normalized to that of
the incident plane wave as a function of the Courant number
(
). It is evident that the field that escapes from the
Huygens’ box is higher for the ADI–FDTD simulation with
the currents evaluated at
and
in the first
and second subiteration, respectively, compared to the case
with all the currents evaluated at
, and increasingly so

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 3, 2004

Gaussian pulse. The simulation is conducted with a temporal
sampling resolution of 21 time steps per period at the maximum
decay point in the pulse specfrequency (defined as the
trum) and a spatial grid resolution of 60 cells per wavelength at
that frequency, which corresponds to a Courant number of 4.9.
Again the results obtained with the currents evaluated at
in both subiterations are more accurate than those obtained with
the currents evaluated at different discrete time values in the two
subiterations.
V. CONCLUSION

Fig. 2. Square root of the average electric energy density computed at a grid
point outside the Huygens’ box, normalized to that of the incident plane wave
as a function of the Courant number in the ADI-FDTD simulation.

Using a systematic approach to build the ADI–FDTD procedure as a perturbation of the Crank–Nicolson scheme, we
have obtained a new method of implementing current sources in
ADI–FDTD. A theoretical analysis and numerical experiments
demonstrate that this approach offers improved accuracy relative to previously reported current source formulations.
REFERENCES

Fig. 3. Electric field computed at a point inside a Huygens’ box, generated by
equivalent currents corresponding to a dipole placed outside the box.

with the time increment. We note that the increase in the error
(a nonzero field outside the Huygen’s box) can be attributed
in part to the intrinsic numerical dispersion error which also
increases with the time increment.
The results of a second numerical experiment are shown in
Fig. 3. Here, we plot the –component of the electric field generated at a point inside a Huygens’ box by the equivalent currents
corresponding to a dipole placed outside the box. The dipole
is oriented along the –axis and is excited with a differentiated

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