A Simplified Novel Link for A Simplified Stability Analysis of
Finite Difference Time Domain Method
OSMAN SAID BISKIN, SERKAN AKSOY
Electronics Engineering Department,
Gebze Technical University,
Gebze, Kocaeli,
TURKEY
Abstract: - Numerical stability and numerical dispersion analyses are critical subjects for Finite Difference
Time Domain (FDTD) method. To perform these analyses, first of all, an equivalency of the FDTD numerical
dispersion equation for Maxwell’s equations and wave equation is proven in this study. Then, based on those
calculations, a simplified version of a novel link is developed. Using this simplified version, a stability criterion
and an amplification factor of the FDTD method are more easily extracted. Therefore, the FDTD stability
analysis becomes simpler. The theoretical findings are validated by a numerical example of a late time
simulation interval in the FDTD method. In particular, the effect of a hard FDTD source and a soft FDTD
source on the growth (amplification) factor is also investigated.
Key-Words: - FDTD method, stability analysis, stability criterion, amplification factor, numerical dispersion.
Received: April 11, 2022. Revised: April 6, 2023. Accepted: May 15, 2023. Published: June 21, 2023.
1 Introduction
Finite Difference Time Domain (FDTD) is a
popular and effective numerical method for the
solution of complex realistic electromagnetic
problems, [1], [2]. Therefore, investigations on
numerical analyses of the FDTD method are a
valuable concern that can be formulated in two
folds: numerical stability analysis and numerical
dispersion analysis, [3]. To perform these two
analyses, first of all, a numerical dispersion equation
(NDE) of the FDTD method must be extracted. This
can be performed in two different ways using
Maxwell’s equations (ME) or wave equation (WE).
If the NDEs are different for ME and WE, the
numerical analyses of any numerical methods in
electromagnetics will inherently differ for ME and
WE, [3]. This makes the numerical analyses more
complex and tiresome. In this sense, one of the
important examples of the numerical time domain
methods is the Pseudo Spectral Time Domain
(PSTD) method. The numerical analyses of the
PSTD show that the PSTD method behaves
differently in the case of ME and WE. This is due to
the eigenvalues of WE PSTD having a second-order
spatial differentiation matrix compared to ME PSTD
having a first-order spatial differentiation matrix
that is closer to the physical models. Therefore, WE
PSTD is more robust to the numerical deficiencies
rather than ME PSTD, [4], [5]. For this reason, in
this paper, this concept is investigated especially for
the FDTD method. First of all, an equivalency
(unification) of the FDTD NDE for ME and WE are
proven. Then, based on those calculations, a
simplified version of a novel link approach is
developed. Thus, a stability criterion and an
amplification factor of the FDTD method are more
easily extracted. This leads to a more simple
numerical analysis of the FDTD method. Finally,
the theoretical findings are validated by a numerical
FDTD example at the late times.
The rest of this article is organized as follows.
The equivalency of the FDTD NDE for ME and WE
is proven in Section I. In Section II, the details for a
complex-frequency approach and a novel link
approach (classical) are revisited. In Section III, the
details for the extraction of a simplified version of
the novel link are given. In Section IV, the
theoretical findings are validated by a numerical
example of a late-time FDTD simulation result. In
particular, the effect of a hard FDTD source and a
soft FDTD source on the growth (amplification)
factor is also investigated. In Section V, conclusions
deducted from the theoretical findings and the
numerical results are discussed.
2 Fundamentals
The numerical analysis of the FDTD method is
intensively investigated by classical methods of
The matrix eigenvalue method,
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The energy method,
The von Neumann (Fourier) method.
The matrix eigenvalue and the energy method
require cumbersome calculations. Therefore, the
most common approach to analyze the stability of
the FDTD method is von Neumann (or Fourier)
method. It is based on a decomposition of the fields
into a discrete complex spatial (position) function
and a discrete real-time function. The first one is
represented by an expansion of an exponential
Fourier series. In the second one, the discrete-time
function satisfies a quadratic equation. Then, the
numerical stability results in a unity-or-less growth
factor of the time function by evaluating the root
locations of its quadratic equation. However, the
rigorous (exact) stability criterion of the FDTD
method cannot be found in this way, [1], [2].
Therefore, some alternative approaches (yet
rigorous) are proposed. They are based on the
numerical analysis of one of the extracted
parameters in the FDTD NDE. In this sense, three
different approaches are
The complex-frequency approach based on a
complex-valued ,
The novel link approach based on a complex-
valued ,
The simplified novel link approach is based
on an equivalency of the NDE for ME and
WE.
In this section, first of all, the complex-frequency
approach and the novel link approach are aptly and
briefly revisited. These two approaches have already
known in the literature, [2], [3]. Then, the simplified
novel link approach is presented in detail. The
novelty of this paper is lying on the third approach.
In order to gain further physical insight and sake
for simplicity, let us consider the one-dimensional
(1D) FDTD NDE for WE as
(1)
In a classical view, the numerical dispersion
analysis of the FDTD method is based on the
extraction of a complex-valued numerical wave
number
from the 1D NDE as
(2)
where , and are considered real-
valued numbers. , is the wavelength at
the given operating frequency () and
defines the grid resolution. In this step, an
alternative compact form of also can be
formulated that
(3)
Now, a transitional value for at limiting
cases of is found as
(4)
where it is worth noting that the numerical
dispersion analysis is based on the evaluations of
only for its real value. The details for this
analysis are given in [2]. Therefore, it is not
repeated here.
For the FDTD stability analysis, the application
of the methods based on the parameter extraction
mentality is similar to the way the extraction of .
However, the extracted (or ) from the NDE is
especially evaluated instead of in the stability
analysis. In the complex-frequency approach (the
first way), and are considered complex-valued
numbers for the numerical dispersion and stability
analyses, respectively. However, in the novel link
approach (the second way), and are
considered complex-valued numbers for the
numerical dispersion and the stability analyses,
respectively. This gives a chance to evaluate the
NDE in different manners.
2.1 The Complex Frequency Approach
This approach is also known as a complex
wavenumber method and is based on the extraction
of the angular frequency from the NDE, [2].
Accordingly, over the discrete wave
is extracted from the NDE as
(5)
where , and are considered real-valued
numbers. The main concept of the complex-
frequency approach in the sense of the input
parameter (real-valued positive numbers of and
) and the output parameter (the complex-valued
number of ) is shown in Fig.1. The complex-
valued number nature of originates from the
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behavior of the function its argument is
given in (5).
Fig. 1: The main concept of the complex-frequency
approach.
Upon close scrutiny of the equation, and
are found as
(6)
where the stability and the instability correspond to
the case of and , respectively
(). Here, it is worth noting that has
complex values in the unstable region.
Now, considering being a real-valued number
() corresponding to a real-valued number
range of , the exact FDTD stability
condition is formulated as
(7)
and considering being a complex-valued number
(), the amplification factor
corresponding to the FDTD instability () is
found to be
(8)
where specially for known as a
magic time step, leads to the stable
algorithm.
Two main disadvantages of the complex
frequency approach are
must be thought of as a complex-valued
number as . This causes a loss
of physical insight for the operational (source)
frequency and leads to controversial
consideration in the time-domain solutions
since is a real-valued input
parameter in reality as not a system (model)
parameter in the FDTD method.
It does not have the ability to perform the
numerical dispersion analysis and the stability
analysis simultaneously. The stability analysis
and the dispersion analysis have to be
independently performed in this way. Let us
explain this situation in detail as is
assumed to be a complex-valued number
while is assumed to be a real-valued
number in the numerical dispersion analysis
whereas is assumed to be a complex-valued
number while is assumed to be a real-
valued number in the stability analysis. It
means that two analyses must be performed,
independently. There is no link between these
two numerical analyses. All these
disadvantages are resolved by proposing a
novel link approach in [3].
2.2 The Novel Link Approach
This approach is based on the consideration of a
complex-valued number of a discrete unit time step
as while keeping is a real-valued
positive number not as before the complex-valued
number of in the complex-frequency
approach for the stability analysis of the FDTD
method, [2]. This is a more reasonable way since the
complex-valued causes in loss of physical insight
for the operational (source) frequency in the time
domain. Moreover, is an input parameter (not a
model parameter of the system), and its values are
already known from the beginning that it must be a
real-valued number rather than the complex-valued
number due to its reality. The novel link approach
resolves this conflict by accepting as the real-
valued number. The main concept of the novel link
approach as input (real-valued positive numbers of
and ) and an output parameter (a complex-
valued number of ) is shown in Fig.2.
Fig. 2: The main concept of the novel link approach.
The novel link approach is based on the
evaluation of that can be a real-valued
number or a complex-valued number. This way is
completely different from the previous technique
since is never used in the complex frequency
approach. In fact, is extracted from the
numerical analysis that is based on the extraction of
. It means that a link is constructed between the
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numerical dispersion analysis and the stability
analysis in the novel link approach. To summarize
this method, let us revisit as
(9)
where considering being a real-valued
number, the exact FDTD stability condition is
(10)
where has the real values in the
range of function.
On the other hand, considering being a
complex-valued number, becomes
(11)
where is a real-valued number () and
defines the time resolution, [3]. By analyzing a
complex-valued number in detail,
and can be calculated that
(12)
By using , the amplification factor
corresponding to the FDTD instability () is
(13)
where let us set to 1 for evaluation of the unit time
step effect. Then, is obtained as
(14)
This formula is the same as the previously published
one of the complex frequency approaches, [2]. Two
main advantages of the novel link approach are
is considered a real-valued number while
becomes the real- or the complex-valued
number in the analyses. This is more
meaningful for reality since is a system
(model) parameter and its behavior cannot be
predicted and restricted from the beginning
due to the fact that it is unknown, yet.
The dispersion analysis and the stability
analysis can be linked with the real-valued
which gives a chance to unify the numerical
analysis of the FDTD method. It is also worth
noting that the novel link approach does not
base on a simple extraction of from the
NDE. This is not simply possible since is
present two times in the transcendental form
of the NDE. Therefore, a numerical root-
finding technique must be used for the
calculation of . However, this is not a
valuable step for the analytical calculations.
Therefore, it cannot be extracted directly
analytically as opposed to and that they
are present only one time in the NDE. The
critical role of the novel link approach is
shown in Fig.3. Accordingly, the numerical
dispersion analysis and the stability analysis
are unified in the novel link approach when
they must be independently considered in the
classical von Neumann (Fourier) method and
the complex-frequency approach.
Fig. 3: The critical role of the novel link approach.
3 The Simplified Novel Link
Approach
The simplification of the novel link is based on
proving the equivalency of the NDEs for ME and
WE. For this aim, more generally, let us consider
the 3D NDE for Maxwell’s equations
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(15)
and also, the 3D FDTD NDE for the wave equation
(16)
where, at first glance, it seems that they are
completely different from each other. Now,
considering a trigonometric relation of
(17)
where or and
applying the following relation for the NDE of ME
(18)
(19)
and substituting them into the ME NDE with a
multiplication of minus two, it becomes
(20)
where it is clear that this form extracted from ME is
exactly equal (unified) to the NDE of WE without
any approximation. They are exactly the same
equations. This makes it possible to remove a
complexity in which the same and unique numerical
analysis is valid and enough for ME or WE.
Extracting the equivalency of the 2D and 1D NDEs
of ME and WE is a straightforward job. Therefore,
it is not shown here. Using this idea, a novel link
proposed in the previous section can be simplified
as shown in the next section. For this aim, first, to
gain further physical insight and to sake for
simplicity, let us again consider the NDE for the 1D
Maxwell’s equations as ()
(21)
where it is necessary to remember that this form of
the ME NDE is equivalent to the WE NDE.
However, the WE NDE (in the form of ) is
used only for the stability analysis in the literature.
Now, for the first time, the ME NDE proving its
equality to the WE NDE is used for the stability
analysis that enables us to find out the simplified
novel link. The simplification of the novel link is
based on using function instead of
function since the equality of the NDE
between ME and WE gives this opportunity. For
further progress, let us extract again from
(2121) given
(22)
where after some mathematical steps, it becomes
()
(23)
Now, evaluating the argument of this simplified
version of at the limit values, being a
real-valued positive number
(24)
where the only limit value is used in the
numerical analysis since the limit value leads to
the negative . The details are given in Appendix I.
Here, this form of is inherently much simpler
than the previous ones based on the WE NDE since
the argument of is only rather than in
the novel link approach.
Now, first of all, consider a region that
in the denominator of takes only the real-
valued positive values as ( must be a
positive valued number because and
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cannot have negative values, individually).
Importantly, this also corresponds to the causality
principle. Then, the exact FDTD stability condition
is extracted as follows
(25)
where must be in the interval of for the
stable solution. In this way, the extraction is
performed in an easier way than the previous two
methods given in the earlier sections.
Second, let us extract the amplification factor of
the FDTD method. Considering being a
complex-valued number, it can be proven that, [7]
(26)
then, becomes
(27)
Now, reconsidering over the parameters of
and as [3]
(28)
where shows the limit value of . Then,
(29)
using this relation, the discrete unit time step is
formulated as
(30)
where is assumed to be a real-valued number as
an input parameter. Upon close scrutiny of the
equation, and are found to be
(31)
where it is clearly figured out that the complex-
valued number is possible. Here, two for the
novel link and the simplified novel link approaches
are already equal to each other. At first glance, it
seems that is different from the previous
extraction of the novel link approach. However,
their equivalency is proven in Appendix II.
Now, the amplification factor () over
corresponding to the FDTD instability () is
extracted by a discrete plane-wave substitution as
(32)
where let us set to 1 for evaluation of the unit time
step effect. Here, it is clear that the same
amplification factor is also extracted in an easier
way without conversion of to as in the novel
link approach. It is directly formulated as a function
of . Especially, the lower limit of the Nyquist
criterion corresponding to is the
worst numerical case of the FDTD method.
In fact, the stability and the instability occur in
the case of and , respectively.
Particularly, the case leads to a numerically
dispersionless solution, corresponding to
known as a magic time step. This also
shows the validity of the simplified novel link
analysis for ME and WE.
4 Numerical Example
A 1D problem in a lossless simple medium is
numerically solved for validation of the theoretical
findings over the amplification factor. The two
FDTD solutions are obtained by developing two
independent FDTD codes for WE and ME. The
parameters of the problem are listed in Table 1.
In the two numerical experiments, the values of
and are set to 2 () and 1.0005,
respectively. The FDTD update equations for the 1D
ME and the 1D WE are well known and can be
found in [1], [2]. Therefore, it is not repeated, here.
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Table 1. The parameters of the problem
The FDTD unit space step ()
0.15 m
The FDTD unit time step ()
0.5 ns
The number of unit cells ()
33333
The problem space ()
5000 m
The operational frequency (f)
1 GHz
Fig. 4a: Comparison of the FDTD amplification
factors between the analytical (formulated) solution,
the ME FDTD solution, and the WE FDTD solution
(the hard source case).
A hard (or a soft) point monochromatic source is
located independently in the middle of the problem
space. An observation point is positioned close to
the source point. In particular, the effect of the hard
FDTD source and the soft FDTD source on the
amplification factor is also investigated.
Accordingly, the comparison of the analytical
(formulated) amplification factor and the
independently FDTD calculated amplification
factors for ME and WE are shown for the hard
FDTD source and the soft FDTD source in Fig.4a
and Fig.4b, respectively.
The numerical results show that
- the two FDTD solutions for ME and WE give the
exactly same results as it is expected from the
proven analytical results in the previous subsections.
- the numerically calculated amplification factors
converge to the analytically (formulated) calculated
ones. They are in good agreement with the steady-
state regime. This is due to the fact that the obtained
analytical solutions are valid at the steady state
regime since the discrete plane wave representation
is used in the analyses.
- the convergence speed of the hard source is slower
than the convergence speed of the soft source.
Since it is shown in the previous chapter that the
analytical formulations of the FDTD amplification
factor for ME and WE are the same, one formula is
enough for all the analytical calculations. On the
other hand, its numerical calculations cannot be
simply performed by a solution of the one FDTD
equation. This is due to the fact that ME and WE
have different orders and different codes. The main
difference is that ME is the first-order equation
while WE is the second-order equation.
Fig. 4b: Comparison of the FDTD amplification
factors between the analytical (formulated) solution,
the ME FDTD solution, and the WE FDTD solution
(the soft source case).
Here, an important example is worthy of mention
that the order difference has a strong effect on the
numerical behavior of the method such as the
Pseudo Spectral Time Domain (PSTD) method, [5],
[6].
5 Conclusion
In this study, a simplified version of the previously
proposed novel link is formulated. This yields the
simplified stability analysis of the FDTD method.
The formulation is based on the equivalency of the
ME FDTD and the WE FDTD numerical dispersion
equations. Thus, the exact stability criterion and the
amplification factor of the FDTD method are more
easily extracted by proving their equivalency. The
theoretical findings are validated by the two 1D
numerical examples (independently for ME and
WE) of the late time simulation interval in the
FDTD method. In particular, the effect of the hard
source and the soft source on the FDTD growth
(amplification) factor is also investigated.
In the simplified version, the operational
frequency is kept again as the real-valued number
which is more physical, and logical and prevents the
loss of physical insight. The unification concept of
the dispersion analysis and the stability analysis is
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also kept in a simpler manner. Thus, all these
calculations ensure a better and simple
understanding of the numerical behavior of the
FDTD method.
The simplified link concept may provide new
openings for a better numerical understanding of the
time domain methods such as the Finite Element
Time Domain (FETD) method, PSTD method, and
so on. In future works, this analysis can be extended
to non-uniform FDTD meshes for more realistic
media such as lossy and dispersive mediums.
6 Appendices
6.1 Appendix I
Considering the numerical wavenumber in the form
(33)
where the argument of has two limit
values as either or . Let us analyze the
limit value by inserting it inside the as
(34)
Using negative argument property of
(35)
transforming to its complex equivalent as
(36)
Now, reconsidering the parameters of
and
(37)
and, using this relation, the discrete unit time step
is formulated as
(38)
Equating , and can be calculated that
(39)
Here, a detailed analysis is given for the limit
value in the argument of . Since this case
leads to the negative , there is no
physical correspondence. From the beginning, it is
declared that must be a real-valued positive
number.
6.2 Appendix II
Let us reconsider and from the novel link
approach ()
(40)
Then, is found to be
(41)
where is the complex-valued
number of the real and the imaginary parts
(42)
Now, let us show and from the
simplified novel link approach
(43)
(44)
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where again the real and the imaginary parts of the
are
(45)
Here, seems to be different from the novel
link. However, one can note that in the novel link’s
does have instead of as an argument. By
taking care of , the equivalency of the two can
be demonstrated as
(46)
(47)
Here, let us define and and
rearrange the above equation
(48)
Then, it becomes
(49)
(50)
Finally, it is clear that this form of is identical
to the form of the novel link as
(51)
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[4] B. Fornberg, A Practical Guide to
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[5] A. Güneş, S. Aksoy, “Long-time instability
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[7] A. Jeffrey and H. H. Dai, “Handbook of
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Contribution of individual authors to the
creation of a scientific article (ghostwriting
policy)
Osman Said Bişkin has carried out the simulation
and the analytical works.
Serkan Aksoy organized the paper and evaluated the
numerical results.
Sources of funding for research presented in a
scientific article or scientific article itself
There is no funding for this research.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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