## 1. Introduction

Present day methodologies for mathematical simulation and computational experiment are generally implemented in electromagnetics through the solution of boundary-value (frequency domain) problems and initial boundary-value (time domain) problems for Maxwell’s equations. Most of the results of this theory concerning open resonators have been obtained by the frequency-domain methods. At the same time, a rich variety of applied problems (analysis of complex electrodynamic structures for the devices of vacuum and solid-state electronics, model synthesis of open dispersive structures for resonant quasi-optics, antenna engineering, and high-power electronics, etc.) can be efficiently solved with the help of more universal time-domain algorithms.

The fact that frequency domain approaches are somewhat limited in such problems is the motivation for this study. Moreover, presently known remedies to the various theoretical difficulties in the theory of non-stationary electromagnetic fields are not always satisfactory for practitioners. Such remedies affect the quality of some model problems and limit the capability of time-domain methods for studying transient and stationary processes. One such difficulty is the appropriate and efficient truncation of the computational domain in so-called open problems, i.e. problems where the computational domain is infinite along one or more spatial coordinates. Also, a number of questions occur when solving far-field problems, and problems involving extended sources or sources located in the far-zone.

In the present work, we address these difficulties for the case of

In contrast to other well-known approximate methods involving truncation of the computational domain (using, for example, Absorbing Boundary Conditions or Perfectly Matched Layers), our constructed solution is exact, and may be computationally implemented in a way that avoids the problem of unpredictable behavior of computational errors for large observation times. The impact of this approach is most significant in cases of resonant wave scattering, where it results in reliable numerical data.

## 2. Formulation of the initial boundary-value problem

In Fig. 1, the cross-section of a model for an open axially-symmetrical (

permittivity

The two-dimensional initial boundary-value problem describing the pulsed axially-symmetrical

where

The domain of analysis

The functions

## 3. Exact absorbing conditions for virtual boundaries in input-output waveguides

Equations

in (1) give the exact absorbing conditions for the outgoing pulsed waves

By using conditions (2), we simplify substantially the model simulating an actual electrodynamic structure: the

In the book (Sirenko et al., 2007), one can find six possible versions of the operators

(nonlocal absorbing conditions) and

(local absorbing conditions). The initial boundary-value problems involved in (5) and (6) with respect to the auxiliary functions

(on the boundaries

(on the boundaries

In (3) to (8) the following designations are used:

Analytical representations for

For

Here

in the circular or coaxial waveguide, respectively.

## 4. Exact radiation conditions for outgoing spherical waves and exact absorbing conditions for the artificial boundary in free space

When constructing the exact absorbing condition for the wave

In the domain

Let us represent the solution

and the following initial boundary-value problem for

Let us solve the Sturm-Liouville problem (15) with respect to

With

Here

where the space-time amplitudes

Our goal now is to derive the exact radiation conditions for space-time amplitudes

Now subject it to the integral transform

where the kernel

Here

Since the ‘signal’

From (24) the simple differential equation for the transforms

In this equation, the values

The last integral is the Hankel transform (Korn & Korn, 1961), which is inverse to itself, and

Taking into account (26), equation (25) can be rewritten in the form

where

and the symbol ‘

Applying the inverse transform (28) to equation (32), we can write

Let us denote

Then from (Gradshteyn & Ryzhik, 2000) we have for

where

where

(Janke et al., 1960), and

Finally, taking into account the relation

By using (19), we arrive at the desired radiation condition:

By passing to the limit

Formula (41) represents the exact absorbing condition on the artificial boundary

## 5. On the equivalence of the initial problem and the problem with a bounded domain of analysis

We have constructed the following closed initial boundary-value problem

where the operator

Here,

By making the following suitable choice of function,

it is possible to show that every term in (43) is nonnegative (Mikhailov, 1976) and therefore

## 6. Far-field zone problem. Extended and remote sources

As we have already mentioned, in contrast to approximate methods based on the use of the Absorbing Boundary Conditions or Perfectly Matched Layers, our approach to the effective truncation of the computational domain is rigorous, which is to say that the original open problem and the modified closed problem are equivalent. This allows one, in particular, to monitor a computational error and obtain reliable information about resonant wave scattering. It is noteworthy that within the limits of this rigorous approach we also obtain, without any additional effort, the solution to the far-field zone problem, namely, of finding the field

and allows one to follow all variations of these amplitudes in an arbitrary region of

given by (40), in turn, enables the variations of the field

It is obvious that the efficiency of the numerical algorithm based on (42) reduces if the support of the function

Let us consider the problem

which differs from the problem (1) only in that the sources

Let the relevant sources generate a field

It follows from (47), (48) that in the domain

and determines there the pulsed electromagnetic wave crossing the artificial boundary

The problems (49) and (14) are qualitatively the same. Therefore, by repeating the transformations of Section 4, we obtain

or, in the operator notations,

## 7. Determination of the incident fields

To implement the algorithms based on the solution of the closed problems (42), (51), the values of the functions

Here (see also Section 3),

The function

In order to find the functions

Here,

Now, by extending the functions

Equations (54) and (58) completely determine the desired function

## 8. Conclusion

In this paper, a problem of efficient truncation of the computational domain in finite-difference methods is discussed for axially-symmetrical open electrodynamic structures. The original problem describing electromagnetic wave scattering on a compact axially-symmetric structure with feeding waveguides is an initial boundary-value problem formulated in an unbounded domain. The exact absorbing conditions have been derived for a spherical artificial boundary enveloping all sources and scatterers in order to truncate the computational domain and replace the original open problem by an equivalent closed one. The constructed solution has been generalized to the case of extended and remote field sources. The analytical representation for the operators converting the near-zone fields into the far-zone fields has been also derived.

We would like to make the following observation about our approach.

In our description, the waveguide

The choice of the parameters

The function

where

The standard discretization of the closed problems (42), (51) by the finite difference method using a uniform rectangular mesh attached to coordinates

As opposed to the well-known approximate boundary conditions standardly utilized by finite-difference methods, the conditions derived in this paper are exact by construction and do not introduce an additional error into the finite-difference algorithm. This advantage is especially valuable in resonant situations, where numerical simulation requires large running time and the computational errors may grow unpredictably if an open problem is replaced by an insufficiently accurate closed problem.