Application of the full reduction technique for solution of equations with vector form non-linearity
We consider making use of the full reduction algorithm for solving the equations with a vector non-linearity. The solutions of such the equations describe the planetary scale non-linear vortex structures of the Earth atmosphere, ionosphere and magnetosphere. We present the modification of full reduc...
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Головна астрономічна обсерваторія НАН України
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| Cite this: | Application of the full reduction technique for solution of equations with vector form non-linearity / D.A. Saliuk // Advances in Astronomy and Space Physics. — 2013. — Т. 3., вип. 2. — С. 98-101. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1196212017-06-08T03:07:34Z Application of the full reduction technique for solution of equations with vector form non-linearity Saliuk, D.A. We consider making use of the full reduction algorithm for solving the equations with a vector non-linearity. The solutions of such the equations describe the planetary scale non-linear vortex structures of the Earth atmosphere, ionosphere and magnetosphere. We present the modification of full reduction technique for Charney-Obukhov equation with periodic boundary conditions. This technique allows to reduce significantly calculation time and to apply much more detailed spatial grid for studying non-linear processes in the near-Earth space. 2013 Article Application of the full reduction technique for solution of equations with vector form non-linearity / D.A. Saliuk // Advances in Astronomy and Space Physics. — 2013. — Т. 3., вип. 2. — С. 98-101. — Бібліогр.: 13 назв. — англ. 2227-1481 http://dspace.nbuv.gov.ua/handle/123456789/119621 en Advances in Astronomy and Space Physics Головна астрономічна обсерваторія НАН України |
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We consider making use of the full reduction algorithm for solving the equations with a vector non-linearity. The solutions of such the equations describe the planetary scale non-linear vortex structures of the Earth atmosphere, ionosphere and magnetosphere. We present the modification of full reduction technique for Charney-Obukhov equation with periodic boundary conditions. This technique allows to reduce significantly calculation time and to apply much more detailed spatial grid for studying non-linear processes in the near-Earth space. |
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Saliuk, D.A. |
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Saliuk, D.A. Application of the full reduction technique for solution of equations with vector form non-linearity Advances in Astronomy and Space Physics |
| author_facet |
Saliuk, D.A. |
| author_sort |
Saliuk, D.A. |
| title |
Application of the full reduction technique for solution of equations with vector form non-linearity |
| title_short |
Application of the full reduction technique for solution of equations with vector form non-linearity |
| title_full |
Application of the full reduction technique for solution of equations with vector form non-linearity |
| title_fullStr |
Application of the full reduction technique for solution of equations with vector form non-linearity |
| title_full_unstemmed |
Application of the full reduction technique for solution of equations with vector form non-linearity |
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application of the full reduction technique for solution of equations with vector form non-linearity |
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Головна астрономічна обсерваторія НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/119621 |
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Application of the full reduction technique for solution of equations with vector form non-linearity / D.A. Saliuk // Advances in Astronomy and Space Physics. — 2013. — Т. 3., вип. 2. — С. 98-101. — Бібліогр.: 13 назв. — англ. |
| series |
Advances in Astronomy and Space Physics |
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AT saliukda applicationofthefullreductiontechniqueforsolutionofequationswithvectorformnonlinearity |
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2025-07-08T16:17:02Z |
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2025-07-08T16:17:02Z |
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1837096159728893952 |
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Application of the full reduction technique for solution
of equations with vector form non-linearity
D.A. Saliuk∗
Advances in Astronomy and Space Physics, 3, 98-101 (2013)
© D.A. Saliuk, 2013
Taras Shevchenko National University of Kyiv, Glushkova ave. 4, 03127, Kyiv, Ukraine
We consider making use of the full reduction algorithm for solving the equations with a vector non-linearity. The
solutions of such the equations describe the planetary scale non-linear vortex structures of the Earth atmosphere,
ionosphere and magnetosphere. We present the modi�cation of full reduction technique for Charney-Obukhov
equation with periodic boundary conditions. This technique allows to reduce signi�cantly calculation time and to
apply much more detailed spatial grid for studying non-linear processes in the near-Earth space.
Key words: experimental and mathematical techniques, numerical simulation studies
introduction
In many cases, after an analytical study of the
non-linear system of di�erential equations, there is
a need to be sure in the correctness of the se-
lected small parameters and obtained partial solu-
tions. Numerical integration of di�erential equations
allows to study the in�uence of small perturbations
in the system. For the description of large-scale
wave structures of Rossby type that are observed
in the atmosphere-ionosphere of the Earth, one uses
the Charney-Obukhov equation [1, 2, 4, 6, 7, 8, 9].
Also in a strong magnetic �eld of plasma a vortex
can exist, similar to Rossby vortices observed in �u-
ids in rotating systems [11, 12, 13]. This analogy
comes from the fact that the form of Hasegawa-
Mima equation for non-linear drift waves in a plasma
fully coincides with the Charney-Obukhov equa-
tion (Charney-Obukhov equation are written for the
stream functions, and the Hasegawa-Mima equation
for the perturbed plasma potential). These equa-
tions contain a vector non-linearity in the form of
Poisson brackets. Let us consider the numerical
scheme to study the dynamics of initial perturba-
tions in the physical system described by Charney-
Obukhov equation:
∂ (∆ψ − ψ)
∂t
+ β
∂ψ
∂y
+ ψ
∂ψ
∂y
+ {ψ,∆ψ} = 0.
We consider the vector non-linearity impact onto
evolution of the vortex structures in the form of Pois-
son brackets. The main problem for numerical in-
tegration of such a system is numerical instability
in the calculation of the �nite di�erence approxima-
tions of the main di�erential equations. That causes
the explosive growth of the kinetic energy of the sys-
tem. The e�ects of this instability are discussed in
detail in [5]. To avoid these e�ects we use the con-
servative form of non-linear vector operators.
application of
full reduction technique
In order to avoid the in�uence of the estimated
e�ects of instability in long-term integration we have
applied numerical methods based on the approxi-
mate representation of vector operations in the �nite
approximation derived from the condition of momen-
tum, kinetic energy, and vorticity conservation in the
system [5]. The equation is considered in a coordi-
nate system where axes x, y with constant step h de-
note distance in space, and z-axis is the amplitude
perturbation. Di�erential equation is re-written in
a form of the system of di�erential equations with
lower order. The calculation has been done in two
steps: numerical integration of next level of Z func-
tion making use of Runge-Kutta 4th order schema
(Z = ∆ψ − ψ), and after obtaining of Zn+1 calcu-
lated on a grid ψn+1. For the calculation of Zn+1 we
use the Arakawa approximation of the second-order
accuracy [5]:
{~a,~b} = − [(bi,j−1 + bi+1,j−1 − bi,j+1 − bi+1,j+1)×
(ai+1,j + ai,j)+
+(bi−1,j−1+ bi,j−1− bi−1,j+1− bi,j+1)(ai,j +ai−1,j)+
+(bi+1,j−1+bi+1,j+1−bi−1,j−bi−1,j+1)(ai,j+1+ai,j)+
+(bi+1,j−1+ bi+1,j − bi−1,j−1− bi−1,j)(ai,j +ai,j−1)+
+ (bi+1,j − bi,j+1)(ai+1,j+1 − ai,j)+
+ (bi+1,j − bi,j+1)(ai+1,j+1 − ai,j)+
+ (bi+1,j − bi,j+1)(ai+1,j+1 − ai,j)+
+(bi+1,j − bi,j+1)(ai+1,j+1 − ai,j)] /
[
12h2
]
,
∗dima.ubf@gmail.com
98
Advances in Astronomy and Space Physics D.A. Saliuk
{~a,~b} = − [(bi,j−1 + bi+1,j−1 − bi,j+1 − bi+1,j+1)×
× (ai+1,j + ai,j)+
+(bi−1,j−1+ bi,j−1− bi−1,j+1− bi,j+1)(ai,j +ai−1,j)+
+(bi+1,j−1+bi+1,j+1−bi−1,j−bi−1,j+1)(ai,j+1+ai,j)+
+(bi+1,j−1+ bi+1,j − bi−1,j−1− bi−1,j)(ai,j +ai,j−1)+
+ (bi+1,j − bi,j+1)(ai+1,j+1 + ai,j)+
+ (bi+1,j − bi,j+1)(ai+1,j+1 + ai,j)+
+ (bi+1,j − bi,j+1)(ai+1,j+1 + ai,j)+
+(bi+1,j − bi,j+1)(ai+1,j+1 + ai,j)] /
[
12h2
]
.
Superscript shows the number of the temporary
layer, subscript i shows the x- and y- coordinates.
The system of equations is modi�ed for periodic
boundary conditions at the edges of the grid.
Here is the system of equations ∆ψ − ψ = Z(x)
on a rectangular grid ω̄ = {xij = (ih1, jh2) ∈ G, 0 ≤
i ≤ M, 0 ≤ j ≤ N, l1 = Mh1, l2 = Nh2}, with
boundary γ, introduced in rectangle G = {0 ≤ xα ≤
lα, α = 1, 2}, that can be presented as the system
of vector equations of a special form [10]. In the
operator form on the grid, the system has a form:{
ψx̄1x2
+ ψx1x̄2
− ψ = −Z(x), x ∈ ω,
ψ(x) = g(x), x ∈ γ,
, (1)
where
ψx̄1x2
=
1
h21
[ψ(i+ 1, j)− 2ψ(i, j) + ψ(i− 1, j)] ,
ψx1x̄2
=
1
h21
[ψ(i, j + 1)− 2ψ(i, j) + ψ(i, j − 1)] ,
ψ(xij) = ψ(i, j).
Now we need to turn the scheme (1). To do this, we
multiply (1) on (−h22) and write out the di�erence
derivative of ψx̄1x2
by points. Now, let ~Ψj be the
vector of M − 1 dimensions, components of which
are the values of the function ψ(i, j) on the internal
nodes of the grid ω̄ on j-th line,
~Ψj = {ψ(1, j), ψ(2, j), . . . , ψ(M − 1, j)}, 0 ≤ j ≤ N,
and ~Fj is a vector with M − 1 dimension:
~Fj = {h22ϕ̄(1, j), h22ϕ(2, j), . . . , h22ϕ(M − 2, j),
h22ϕ̄(M − 1, j)}, 0 ≤ j ≤ N,
~Fj = {g(1, j), g(2, j), . . . , g(M − 1, j)}, j = 0, N.
Let us de�ne the matrix Ĉ of (M − 1) × (M − 1)
dimensions as follows:
Ĉ ~V = {Λv(1),Λv(2), . . . ,Λv(M − 1)}~V =
= {v(1), v(2), . . . , v(M − 1)},
where di�erential operator Λ is introduced in a form:
Λv(i) = 2v(i)− h22vx̄1x2
(i), 1 ≤ i ≤M − 1,
v(0) = v(M) = 0.
Matrix Ĉ has the form:
Ĉ =
∣∣∣∣∣∣∣∣
2(1 + α) −α 0 · · · 0
−α 2(1 + α) −α · · ·
0 −α 2(1 + α) · · · 0
· · · −α 2(1 + α) −α
0 0 · · · −α 2(1 + α)
∣∣∣∣∣∣∣∣ ,
where α =
h22
h21
. In the case of 1 + α > α, the matrix
Ĉ is non-degenerate.
Now we can proceed to the system of vector equa-
tions of a special form with constant coe�cients: −~Ψj−1 + Ĉ ~Ψj − ~Ψj+1 = ~Fj , 1 ≤ j ≤ N − 1
~Ψ0 = ~F0,
~ΨN = ~FN .
Regular methods for the solution of such a system
are given in detail in [10]. Making use of the results
for the particular case of equations with constant co-
e�cients we �nd the solution in a rectangular grid.
The idea of full reduction technique is to exclude the
unknown ~Ψj with odd j from the equations, then
with j, multiples of two, then four, etc. Each new
step of the exclusion process reduces the number of
unknown variables, and if N is the degree of two,
i. e. N = 2n, then as a result the values of ~ΨN/2 are
obtained. The reverse run of algorithm provides cal-
culation of the values ~Ψj with numbers j, multiple
of N/4, then N/8, N/16, etc. As a result, we obtain
the following equations for calculating ~Ψj through
the vectors ~p and ~S:
S
(k−1)
j =
2k−1∏
l=1
αl,k−1C
−1
l,k−1
(
~p
(k−1)
j−2k−1 + ~p
(k−1)
j+2k−1
)
,
~p
(k−1)
j = 0, 5
(
~p
(k−1)
j + ~S
(k−1)
j
)
, ~p
(0)
j ≡ ~Fj ,(
j = 2k, 2 · 2k, 3 · 2k . . . N − 2k, k = 1, 2, . . . , n− 1
)
;
~Ψj =
2k−1∑
l=1
C−1
l,k−1
[
~p
(k−1)
j +
+αl,k−1
(
~Ψ
(k−1)
j−2k−1 + ~Ψ
(k−1)
j+2k−1
)]
,
99
Advances in Astronomy and Space Physics D.A. Saliuk
Fig. 1: The scheme of the transformation of a rectangular grid into the torus with periodic
boundary conditions.
~Ψ0 = ~F0, ~ΨN = ~FN ,(
j = 2k, 3 · 2k, 5 · 2k..N − 2k, k = n, n− 1, . . . , 1
)
,
where
C−1
l,k−1 = C − 2 cos
[
(2l − 1)π
2k
]
E αl,k−1 =
=
(−1)l+1
2k−1
sin
[
(2l − 1)π
2k
]
.
The obtained formulae describe the application of
the full reduction technique with just vector addi-
tion, vector multiplication by number and inversion
of matrices. If Ĉ is a three-diagonal matrix, then
any Cl,k−1 will be also three-diagonal as well.
numerical simulations
Applying the algorithm of full reduction we de-
veloped the modi�cation for periodic boundary con-
ditions. Algorithms of numerical integration have
been implemented in the IDL programming language
on the grid 4096 × 1000 with a step of 0.1 in time
and in space.
The type of solution depends on the ratio of the
linear and non-linear terms of equation. In a linear
case the initial vortex disturbance solution is unsta-
ble and rapidly decays into linear waves. In a non-
linear system the vortex solution is stable and weekly
attenuates during the drift (Fig. 2).
conclusions
We present the modi�cation of the full reduc-
tion technique (periodic boundary conditions and
additional terms in equation) and its application for
study the dynamics of the system with vector non-
linearity. The algorithm adapted for a rectangular
grid of integration with periodic boundary conditions
was developed and implemented. The dynamics of
the initial perturbation in a system with vector non-
linearity using the methods of numerical integration
was studied. The integration was carried out on a
grid with a constant step. Acceleration allows to
provide the numerical calculations on the grid size
1000× 1000 (2-3 hours).
references
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linevskyG.P. 2006, Rep. National Academy of Sciences
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[2] AgapitovA.V., GrytsaiA.V. & SaliukD.A. 2011, Space
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100
Advances in Astronomy and Space Physics D.A. Saliuk
Fig. 2: Dynamics of the initial perturbation in a form of the monopole vortex with the Gaussian
potential. a) Dynamics of the linear system (the initial structure decays to linear waves). b)
Dynamics of the nonlinear system (nonlinear vortex solution is stable).
101
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