=Paper=
{{Paper
|id=Vol-1900/paper8
|storemode=property
|title=The elaboration of numerical simulation error light pulse propagation in a waveguide of circular cross-section
|pdfUrl=https://ceur-ws.org/Vol-1900/paper8.pdf
|volume=Vol-1900
|authors=Alexandr A. Degtuarev,Alexandra V. Kukleva
}}
==The elaboration of numerical simulation error light pulse propagation in a waveguide of circular cross-section ==
https://ceur-ws.org/Vol-1900/paper8.pdf
The elaboration of numerical simulation error light pulse propagation in a
waveguide of circular cross-section
A.A. Degtuarev1, A.V. Kukleva1
1
Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia
Abstract
We considered the problem of estimating the error in the solution of the wave equation recorded using infinite series Fourier-Bessel. The
algorithm that adjusts the number of elements in a partial sum of infinite series, based on the assessment of the series balance. The application
of the algorithm made it possible, without loss of accuracy, to substantially reduce the number of summable elements of the series in the
numerical simulation of the light pulse propagation in a circular cross-section.
Keywords: wave equation; Fourier-Bessel series; evaluation of the residual series; numerical simulations; pulse of light; computational
experiment; redundancy of partial sum components
1. Introduction
During the development of an application program for the numerical simulation of a physical process, it is important to
investigate the actual error of the method used on special test cases. As test cases typically use such examples that can be
resolved by an alternative method with high sufficiently precision, allowing to calculate the error of numerical method [1, 2].
This work is devoted to study the error of test value problem for the wave equation describing the propagation process of the
light pulse in a waveguide in circular cross section. To elaboration the error estimate, we used remainder of the Fourier-Bessel.
To check the quality of the balance assessment in the series we used the technique of computational experiment, which allows
determine the degree of redundancy among several elements needed to sum to achieve the necessary precision [3].
In solving problems from numerical simulation propagation of a light pulse in a medium, various mathematical descriptions
of the pulse [4-6]. In this paper, we considered two options describe different degrees of smoothness pulse function.
2. Mathematical model of light pulse propagation in a waveguide of circular cross-section
To describe the process of light pulse propagation we will consider the following boundary value problem:
2 E c 2 2 E 1 E 2 E
2 2 2 , r 0; R , z 0; L , t 0; T ;
t n r r r z 2
E 0, r 0; R , z 0; L ;
t 0
E
0, r 0; R , z 0; L ;
t t 0
E
z 0 r , t , r 0; R , t 0; T ;
E
0, r 0; R , t 0; T ;
z z L
E
r R 0, z 0; L , t 0; T ,
where E is a dielectric field intensity, c is a wave propagation speed in vacuum, n is a refractive index material of the
waveguide, R and L is the radius and length of the waveguide, T is the duration of the dissemination process, (r , t ) is the
function describing the pulse shape.
It is assumed when r R an ideally conducting shell bound the waveguide, and the medium is not perturbed at the initial
instant of time.
Here are the following two variants of kinetic moment:
1 (r, t ) r t sin t , 2 (r , t ) r t sin t sin 2 *t ,
1, t 0; t * ; 2 c 2 c
where t , * , t * is the pulse duration at the entrance of the waveguide, is the
0, t t ; Tt , j
*
length of disturbing wave in vacuum, j a positive integer. 1 (r , t ) a piecewise smooth function at variable t , because
derivative has function jump in t 0 , t t * . Function 2 (r , t ) has the smoothness of a second-order variable t .
3rd International conference “Information Technology and Nanotechnology 2017” 38
Computer Optics and Nanophotonics / A.A. Degtuarev, A.V. Kukleva
3. Exact solution of boundary value problem
Application of the separation variables method [5] allows getting solution of boundary-value problem for the wave equation,
it can be thought of as infinite series Fourier-Bessel. For example, when describing an impulse function 1 (r , t ) and using
r J 0 1r the solution would be:
sin k t ˆ 2 k2 k sin t ˆ 2 2
E r , z, t J 0 (1r ) ck sin k z sin t , if t 0; t * ;
k 0 k k
2 2
a2 t *
E r , z, t J 0 (1r ) sin k z a1 t * cos k t t * sin k t t * , if t t * ; T .
k
k 0
When writing these formulas, we use the following notation:
4 2k 1 c c
1 1 , ck , k , k k2 12 , ˆ 1 ,
R 2k 1 2L n n
sin k t ˆ 2 k2 k sin t ˆ 2 2
a1 t ck sin t ,
k k2 2
cos k t ˆ 2 k2 cos t ˆ 2 2
a2 t ck cos t .
k2 2
Graph of the cross section of a pulse by a plane r 1 m in the process of its propagation in the waveguide has shown in
fig.1.
Fig. 1. Modeling the distribution piecewise smooth impulse in wave conductor, separation r 1 m .
For the case of smooth pulse described by function 2 (r , t ) if * and, r J 0 1r solution of boundary-value
10
problem is as follows:
c t
E r , z , t J 0 (1 r ) k sin k z 0.5a3 t a4 t a5 t sin t sin 2 , if t 0; t ;
*
k 0 k 10
a7 t *
E r , z, t J 0 (1r ) sin k z a6 t * cos k t t * sin k t t * , if t t * ; T .
k
k 0
In the last formulas, we used the following notations:
ˆ 2 2
a3 t 2 sin k t k sin t ,
k2
0.16 2 0.25ˆ 2 4 0.36 2 0.25ˆ 2 6
a4 t 5 4 sin k t 5k sin t , a5 t 5 6 sin k t 5k sin t ,
16 25k
2 2
5 36 2 25k2 5
t
a6 t
ck
k
0.5a t a t a t c sin t sin ,
3
*
4
*
5
*
k
10
2
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Computer Optics and Nanophotonics / A.A. Degtuarev, A.V. Kukleva
t
'
a7 t
k
ck
0.5a3' t * a4' t * a5' t * ck sin t sin 2 ,
10
1 is a root of an equation J 0 R 0 .
The process of propagating a piecewise-smooth pulse has shown in figure 2.
Fig.2. Modeling of smooth pulse in wave conductor, separation r 1 m .
4. Series truncation error control
A computer program simulating the spread of pulse truncation of the infinite series implied above.
N
If we can get an estimate a balance number of E r , z , t u k r , z , t in the form of
k 1
RN u r, z, t N ,
k N 1
k
where N is the positive monotonically decreasing function if N , this assessment can be used to control the
truncation error. To do this, we need only find N ( ) , is the least value N , satisfy the inequality N , and for
N ( )
approximate calculation of function values E ( x, z , t ) use a partial amount EN ( ) en ( x, z, t ).
n 1
In this case, the actual error of the calculated value of a function E at the selected point does not exceed the required level
, that is
fact E E N ( ) RN ( ) ( N ( )) .
For the above two ways to specify the light pulse residues had been received by the relevant rows with the following t * and
w* :
c * 10
w* , t .
5 c
In the case of piecewise smooth impulse, that described function 1 (r , t ) , assessed takes the following form:
8Ln 1.003 2nL 2 ˆ 2
EN r , z, t ,
2 N 1 2 2 4nL
c 3 2N
as for the case of smooth pulse, that described function 2 (r , t ) , assessed takes the following form:
0.16 n 2 L2 2
EN r, z, t .
c 2 3 2 N 1
2
It should be noted that recorded higher truncation error estimates infinite series are uniform for all independent variables.
3rd International conference “Information Technology and Nanotechnology 2017” 40
Computer Optics and Nanophotonics / A.A. Degtuarev, A.V. Kukleva
5. The method of refinement of the number of summable elements of a series using a computational experiment
Proposed evaluation are not ideal because they are using strict inequalities, and also they are uniform for all independent
variables. That is why using of estimates results in adding more elements than is necessary to achieve the required accuracy. In
this case, it is advisable to apply a technique, which reduces the degree of redundancy terms in the partial sum, and in so doing
guarantees the achievement of required accuracy [3].
Let N positive integer, satisfies the inequality N N 1 , where 1 , number N 1 found by the rule described in
paragraph 4. Then for partial amount EN the actual error will satisfy the inequality:
fact N E EN E EN EN EN .
1 1
Changing N within the boundaries N 1 N 1 , find lowest value N 2 , when running the inequality
EN 1 EN 2 , where 2 1 .
For this choice 2 and equity of the previous inequality, the actual error fact ( N ( 2 )) do not exceed value .
Thus, to reduce the number of summands in the partial amount, we must:
1) Specify the number of 1 and then find the value N (1 ) , that the smallest value N , satisfy the inequality
( N ) 1 .
2) Changing a variable N from the value N 1 downward, find the smallest of its value that satisfies the inequality
EN (1 ) EN 2 . The resulting value is N 2 .
3) Changing value with sample spacing 1 and 2 so, to 1 2 , run the steps 1) and 2) again.
4) Of all the values N 2 , obtained in step 3), select the smallest.
As a result of the use of this algorithm, it can be expected that the number of summable elements N 2 in the partial sum
will be reduced significantly as compared with the number of N while maintaining safeguards for accuracy, i.e.
fact N 2 .
In tables 1 and 2 are the results of computational experiments, aimed at reducing the number of summands in partial
amounts. The calculations have been carried out with the following parameters:
tc
1 m, n 1, L 7 m, R 5 m, с 3 1014 m / s, r 1 m, z 1 m, t m .
n
Asked value in increments of the maximum value of the amplitude of the wave.
Table 1. The dependence of the summands number N and N 2 of coordinate t with different values for piecewise smooth impulse.
10-1 10-2 10-3 10-4 10-5
N 131 1019 9844 98079 980434
t , m N 2
0.9 13 48 231 3116 9906
0.999 37 306 1241 6774 26632
0.99999 37 312 3072 35599 126836
1 37 312 3075 37713 377122
1.00001 37 312 3072 35599 126836
10-1 10-2 10-3 10-4 10-5
1.001 37 306 1241 6774 26633
1.1 16 68 320 3119 9906
1.7 19 34 124 1286 4086
2.5 15 32 96 928 984
4 13 25 66 612 643
5.1 16 30 75 649 1436
5.9 17 28 324 3116 3258
5.999 47 355 1262 6422 9906
5.99999 47 466 4672 35599 26632
6 47 467 4672 37713 126836
6.00001 47 465 4671 35599 377122
3rd International conference “Information Technology and Nanotechnology 2017” 41
Computer Optics and Nanophotonics / A.A. Degtuarev, A.V. Kukleva
6.001 47 383 1461 6423 126836
6.1 17 30 360 3119 26634
Table 2. The dependence of the summands number N and N 2 of coordinate t with different values for smooth pulse.
10 -2
10-3 10-4 10-5 10-6
N 21 36 113 357 1128
t , m N 2
0.9 15 18 28 62 132
0.999 15 18 28 61 136
0.99999 14 17 28 62 126
1 15 18 27 67 141
1.000001 10 21 37 91 186
1.001 10 22 42 101 211
1.1 15 17 33 61 132
1.7 10 17 37 81 181
2.5 13 15 26 67 146
4 15 22 46 101 216
From the table it can be seen that the number of summands, using uniform assessments for the respective series truncation
allows you to get only the rough partial sums of lengths. These values are repeatedly exceed the values obtained from the
application of the above algorithm. As can be seen from table 1, to calculate the tension of the electric field in the foreground
and background areas of wave fronts requires a much larger number of terms, for example, in the range 1.7 m t 5.1 m
order enough 4086 parts to achieve precision 10-5, while in the range 0.9 m t 1.1 m we want 377122 parts.This increase
in the number of summands is a consequence of the weak function breaks 1 (r , t ) , significantly slowing down the convergence
of series. For the case of smooth pulse, function description 2 (r , t ) the uneven distribution of values N 2 for different t
turns out to be negligible.
6. Conclusion
Developed and implemented programmatically algorithm provides adjustment of the partial sums length of infinite series,
obtained in the course of solving boundary value problem for the wave equation. For practical application of the algorithm, it is
of fundamental importance to first obtain an upper estimate for the remainder of the Fourier series that determines the solution of
the boundary value problem.
The application of developed algorithm for specific series that describe the distribution of momentum in circular waveguide
section allowed multiple times (from 3 up to 1500 times and more for Piecewise-smooth momentum and from 2 to 5 times for
the case of smooth pulse) to reduce length of the partial sums of the series.
References
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Mathematics 2012; 1: 1–7.
[2] Degtyarev AA, Kozlova ES. Investigation of accuracy of numerical solution of the one-way Helmholtz equation by method of computational experiment.
Computer Optics 2012; 36(1): 36–45.
[3] Degtyarev AA, Praslova MO. Estimation of the errority of the solution of the wave equation in the problem of modeling the distribution of the light pulse in
the planary waveguide. Proc. of ITNT-2016, Samara, SSAU 2016; 852–859. (in Russian)
[4] Kotlyar VV, Kozlova ES. Simulation of ultrafast 2d light pulse. Computer Optics 2012; 36(2): 158–164.
[5] Kotlyar VV, Kozlova ES. Simmulations of Sommerfeld and Brillouin precursors in the medium with frequency dispersion using numerical method of
solving wave equations. Computer Optics 2013; 37(2): 146–154.
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