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spusyal sovtwaru vor somputynw thu
spusyal vunstyons ov wavu satastrophus
sovtwaru uspusyal para salsular las
vunsyonus uspusyalus tu
satastrovus tu olas
Antruw c. Kryukovsky∗ Tmytry c. Lukyn†
curwuy f. Rowashuv‡
gexeiveyO FAaiAP geviseyO FAaiAP
AxxepteyO FAaiA
Snpuyty os Insormntvon Systrms nnq Pomputrr arpunoyoty9 Russvnn Nrw bnvvrrsvty9
Rnqvo st.9 229 105005 Mospow9 Russvn. R:mnvy: kryukovskyMrnmoyrr.ru
yMospow Instvtutr os Puysvps nnq arpunoyoty 5Stntr bnvvrrsvty)9 Instvtutskvy ynnr F9
1A1D00 Doytopruqny9 Mospow Rrtvon9 Russvn. R:mnvy: yukvnMmnvy.mvpt.ru
zMvsmn qvrrppvon qur/same address as N.S. Kryukovsky. R:mnvy:
rotnpurvsrrtrvMtmnvy.pom
21
22 a.s. kryukovsky { t.s. lukyn { s.v. rowashuv
Avstruwt
The method of ordinary differential equations in the context of
calculating the special functions of wave catastrophes is considered.
Complementary numerical methods and algorithms are described.
The paper shows approaches to accelerate such calculations using
capabilities of modern computing systems. Methods for calculat-
ing the special functions of wave catastrophes are considered in the
framework of parallel computing and distributed systems. The pa-
per covers the development process of special software for calculating
of special functions, questions of portability, extensibility and inter-
operability.
Kyywords: ODE method; special functions of wave catastrophes; nu-
merical methods; algorithms; parallel computing; distributed computing.
fysumyn
Se considera el me´todo de ecuaciones diferenciales ordinarias or-
dinarias en el contexto de calcular funciones especiales de cata´stro-
fes de olas. Se describen me´todos y algoritmos nume´ricos comple-
mentarios. El art´ıculo muestra enfoques para acelerar tales ca´lculos
usando capacidades modernas de sistemas de ca´lculo. Se conside-
ran me´todos para calcular funciones especiales de cata´strofes de olas
en el marco de computacio´n en paralelo y sistemas distribuidos. El
art´ıculo cubre el proceso de desarrollo de software especial para cal-
cular funciones especiales, as´ı como asuntos de portabilidad, exten-
sibilidad e interoperabilidad.
duluvrus wluvy: me´todos de EDO; funciones especiales de cata´strofes
de olas; me´todos nume´ricos; algoritmos; computacio´n en paralelo; com-
putacio´n distribuida.
authymutiws guvjywt Clussiwution: 78A99.
1 IncaoddRcion
The theory of wave catastrophes is an important field, which has a direct
practical importance in areas such as radiolocation, formation of direc-
tional beams in plasma, transfer of data over fiber-optic lines, etc. Meth-
ods of the theory of wave catastrophes allow to go beyond the boundaries
of the known asymptotic methods in the study of electromagnetic wave
propagation.
This article describes approaches to numerical calculation of special
functions using modern computing systems. The software product based
Rev.Mate.Tear.Aplic. ISSN )4(1-2433 (Prinl) 22)--3373 (Online) Vol. 22())2 2){+(, Jan 2()5
somputynw thu spusyal vunstyons ov wavu satastrophus 23
on these methods is planned to be embedded to the information system
”Wave catastrophes in radio physics, acoustics, and quantum mechanics.”
2 O34 mechod
Known methods of the theory of wave catastrophes (for example, the con-
tour method, summing the Taylor series) are difficult to apply for multi-
parameter catastrophes as their complexity increases with the number of
parameters [3, 5]. The method of ordinary differential equations is free
from these disadvantages [4].
ODEs for a rather small number of catastrophes are derived by hand
[7], but, in theory, this can be done automatically. The algorithm can be
implemented quite simply using one of the modern packages of symbolic
computation (Maxima or Wolfram Mathematica, for example).
Dynition 1 A vextor xonsisting of v fivve xvtvstrophe spexivl funxtion
vny v pvrt of its rst yirevvtives is v funyvmentvl vextor of this spexivl
funxtion =for exvmpleA funyvmentvl vextor of A+ xvtvstrophe is =F))O
W⃗ = (k; k ); k 2); 
 = 3O (1)
ihe other yerivvtives of v spexivl funxtion vre uniquely expressey in terms
of the funyvmentvl vextor's xomponents fiith the help of linevr vlgewrvix
relvtions resulting from the xvnonixvl system of yifferentivl equvtionsC
Argues that for calculation of the fundamental vector’s components a
system of ordinary differential equations can be derived from the system
of canonical equations. The resulting system can be easy calculated by
numerical methods like Runge-Kutta or Kutta-Merson.
For example, consider the V+ catastrophe (2):
kA3(⃗) =
∫ +∞
−∞
zxp(i(x+ + 2x
2 + )x)) yxO (2)
The following system of canonical equations corresponds to the catas-
trophe described above [4]:
() − 4 U
2
U2U)
− 2i2 U
U)
)I = 0;
(
U2
U2)
− i U
U2
)I = 0O
(3)
Rev.Mate.Tear.Aplic. ISSN )4(1-2433 (Prinl) 22)--3373 (Online) Vol. 22())2 2){+(, Jan 2()5
24 a.s. kryukovsky { t.s. lukyn { s.v. rowashuv
The system of ODEs (4) can be derived from the system of canonical
equations [4, 2]:
yk
y)
= k );
yk
y2
= k 2;
yk )
y)
= ik 2;
yk 2
y)
=
1
4
()k − 2i2k )) ≡ U2); yk
)
y2
= U2);
yk 2
y2
= − i
4
(k + )k
) + 22k
2)O
(4)
The set of initial conditions (5) supplements the system:
k (0) =
1
2
Γ(
1
4
)zxp(
i.
8
);
k )(0) = 0;
k 2(0) =
i
2
Γ(
3
4
)zxp(
i3.
8
)O
(5)
The method of deriving a system of ODEs can be described by the
following steps [4, 6]:
• First of all, it is necessary to select the starting point. It may be
2 = 0 and 2 = 0, for example (for V+).
• After the selection of the starting point for a future system of ODEs
it is needed to determine the components of the vector W⃗ and its
length. This can be done by drawing up a table, each row of which
corresponds to the function itself or one of its first derivatives, and
the columns correspond to the derivatives with respect to each pa-
rameter (the table corresponding to the V+ catastrophe (Table 1)).
• All components of the table are derivable from the system of canon-
ical equations. The table also helps to determine the amount of
fundamental vector’s components.
3 NdmeaiRPl mechodb Pnd Plgoaichmb
The software solution uses the Kutta-Merson method to calculate special
functions with different sets of parameters. This is a method with fourth-
order accuracy and step autocorrection [8].
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somputynw thu spusyal vunstyons ov wavu satastrophus 25
K
K)
K
K2
k (I, I) (I, II)
k) (II, I) (II, II)
k2 (III, I) (III, II)
huvly 1: Table corresponding to the V3 catastrophe.
.
Consider a system of the form:
x˙ = f(t; x)O (6)
To integrate the system following formulas can be applied:
xn+) = xn +
1
2
(k) + 4k, + k5) +d(h
5);
k) =
1
3
hf(tn; xn);
k2 =
1
3
hf(tn +
1
3
h; yn + k));
k+ =
1
3
hf(tn +
1
3
h; yn +
1
2
k) +
1
2
k2);
k, =
1
3
hf(tn +
1
2
h; yn +
3
8
k) +
9
8
k+);
k5 =
1
3
hf(tn + h; yn +
3
2
k) − 9
2
k+ + 6k,)O
(7)
The local truncation error is expressed by the following formula:
 ∼ k) − 9
2
k+ + 4k, − 1
2
k5O (8)
There are several criteria to determine how to change the integration
step. In the current implementation the step is multiplied by 2 if the local
truncation error is less than 5+2E (the specified accuracy of calculation).
If the local truncation error exceeds 5E the step must be devided by 2. In
the other cases the step is not changed.
Studied catastrophes are complex multi-variable systems; to calculate
their special functions and visualize the result as a 3D plot it is needed to
choose parameters which will be changed discretely, other parameters are
assigned to some fixed values.
Rev.Mate.Tear.Aplic. ISSN )4(1-2433 (Prinl) 22)--3373 (Online) Vol. 22())2 2){+(, Jan 2()5
26 a.s. kryukovsky { t.s. lukyn { s.v. rowashuv
Dutu: Initial parameters of SODE
fysult: Set of SODE solutions
initialization;
for Firsevrvm OR binFirstevrvm C C CbvxFirstevrvmA
Firstevrvmhtep do
for hexonyevrvm OR binhexonyevrvm C C CbvxhexonyevrvmA
hexonyevrvmhtep do
prepare the initial vector;
ResultVector := KuttaMersonMethod(InitialVector);
save the resulting vector for future use;
ynd
ynd
Algorithm 1: Sequential special wave function calculation.
The generic sequential algorithm for calculation special functions of
wave catastrophes can be expressed with the simple enough pseudocode
(Algorithm 1).
Usually it is enough to integrate the system from 0.0 to 1.0.
It is easy to note that the system can be integrated in the parallel
environment by the devision of the set containing descrete parameters’
values to a number of subsets. Integration in these parameter spaces
can be performed independently and the results can be merged after the
calculation. Thus, the algorithm showed above fits the parallel solution,
and does not require any change. On the contrary, the set of input values
must be prepared, and each thread in the system must get a separate part
of this set.
The performance gain can be calculated using the formula (9). Assume
c is a number of independent calculating units (cores, processors, network
nodes, etc), time of calculation is proportional to )N :
iplrlllpl = C
ispqupnttll
c
O (9)
4 PoacPbilich Pnd egcenbibilich
One of the main requirements to the software developed in this work
is portability and this requirement is refracted in some design solutions
assumed in the project. C++ is used as a main programming language:
it helps to develop very portable solutions which can be built for many
modern architectures like x86, ARM, MIPS, etc. Graphical interface and
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somputynw thu spusyal vunstyons ov wavu satastrophus 27
OpenGL graphics are drown through the FLTK library (very portable and
tiny GUI toolkit), network communication and multithreading are based
on small self-developed libraries (MS Windows, GNU/Linux and many
other Unixes are supported).
The software uses a common intermediate representation of float point
numbers to communicate between network nodes. Thus, these nodes can
use different internal representations and work together in one computing
network.
To simplify development of extensions the software supports a special
domain specific language. ODEs of wave catastrophes can be described
in the language and used to calculate these catastrophes. A part of this
functionality is still under development.
5 4gPmpleb of eibdPliiPcion
The equation of the catastrophe V+ is already described above (2), Figure
1 shows its module.
Figury 1: V3, module. 1 = −15 O O O+ 15, 2 = −10 O O O+ 10.
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28 a.s. kryukovsky { t.s. lukyn { s.v. rowashuv
Consider the graphical representation of the corner catastrophe V,) (see
Figure 2):
kA4)(); 2; a) =
∫ +∞
(
∫ +∞
(
zxp(i(k)z
2 + ayz + k2y
2 + )y + 2z))yy yzO
(10)
Figury F: V41, phase. 1 = −10 O O O + 10, 2 = −15 O O O + 5, 1 = −1, 2 = 1,
 = 2xos(4 ).
The edge catastrophe K,;2 described by the equation 11 is shown on
figures 3 and 4:
kK4;2(); O O O ; ,; ) =
=
∫ +∞
(
yz
∫ +∞
−∞
yx(i(z2 + x2z + x, + )x+ 2x
2 + +z + ,zx))O (11)
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somputynw thu spusyal vunstyons ov wavu satastrophus 29
Figury 3: K4;2, module. 1 = −8 O O O + 8, 2 = −7 O O O + 7, 3 = −2, 4 = −2,
 = −1.
Figury 4: K4;2, phase. 1 = −8 O O O + 8, 2 = −7 O O O + 7, 3 = −2, 4 = −2,
 = −1.
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30 a.s. kryukovsky { t.s. lukyn { s.v. rowashuv
RefeaenReb
[1] Arnold, V.I. (1992) Cvtvstrophe iheory, 3rd edition. Springer, Berlin.
[2] Dorokhina, T.V.; Kryukovsky, A.S.; Malyshenko, A.B. (2008) “Devel-
opment of numerical algorithms for the calculation and visualization of
wave catastrophes”, goscdU WulletinC bvnvgementA Computing vny
Informvtixs 3 Russian New University, Moscow: 25–47.
[3] Ipatov, E.B.; Kryukovsky, A.S.; Lukin, D.S.; Palkin, E.A. (1986)“Edge
catastrophes and asymptotics”, geports of the Axvyemy of hxienxes,
USSR, FM1(4): 823–827.
[4] Kryukovsky, A.S. (1992)“Method of ordinary differential equations for
computing the special functions of wave catastrophes (SWC)”, DiffrvxB
tion vny eropvgvtion of Elextromvgnetix lvvesO Intervgenxy CollexB
tion of hxientix evpers, Moscow: MIPT: 29–48.
[5] Kryukovsky, A.S.; Lukin, D.S.; Palkin, E.A. (1995) “Uniform asymp-
totics and corner catastrophes”, geports of the gussivn Axvyemy of
hxienxes 341(4): 456–459.
[6] Kryukovsky, A.S. (2013) “The uniform asymptotic theory of edge
and corner wave catastrophes”, Monograph, Russian New University,
Moscow.
[7] Kryukovsky, A.S.; Lukin, D.S. (2003) “The theory of calculating the
reference focal and diffraction electromagnetic fields on the basis of
special functions of wave catastrophes”, iexhnology vny Elextronixs
48(8), Moscow (in Russian): 912–921.
[8] Shampine, L.F. (2005) “Error estimation and control for ODEs”, JourB
nvl of hxientix Computing F5(1): 3–16.
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