Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method
Problem of the plasma simulation area decomposition for Poisson equation solution via PIC method using parallel computing is treated. Modification of the existing method moving to reduction of the computational complexity is discussed. The approach to use different grid spacing for subdomains is p...
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irk-123456789-1153222017-04-03T03:02:28Z Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method Dadyka, D.I. Anisimov, I.O. Basic plasma Problem of the plasma simulation area decomposition for Poisson equation solution via PIC method using parallel computing is treated. Modification of the existing method moving to reduction of the computational complexity is discussed. The approach to use different grid spacing for subdomains is proposed. Method of approximate solution that results in the substantial decrease of the data transmitted is brought forward Рассмотрена проблема декомпозиции области моделирования для уравнения Пуассона методом крупных частиц в ячейках. Предложены пути минимизации вычислительных затрат. Продемонстрирована возможность использования различного шага сеток для подобластей. Предложен метод значительного снижения объёма передаваемых между вычислительными узлами данных. Розглянуто проблему декомпозиції області моделювання для рівняння Пуассона методом крупних частинок у комірках. Запропоновано шляхи мінімізації обчислювальних витрат. Продемонстровано можливість використання різного кроку сіток для підобластей. Запропоновано метод значного зниження об'єму даних, що передаються між обчислювальними вузлами. 2016 Article Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method / D.I. Dadyka, I.O. Anisimov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 81-83. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 02.60.Pn, 52.65.Rr, 52.80.Tn http://dspace.nbuv.gov.ua/handle/123456789/115322 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma Basic plasma |
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Basic plasma Basic plasma Dadyka, D.I. Anisimov, I.O. Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method Вопросы атомной науки и техники |
| description |
Problem of the plasma simulation area decomposition for Poisson equation solution via PIC method using
parallel computing is treated. Modification of the existing method moving to reduction of the computational
complexity is discussed. The approach to use different grid spacing for subdomains is proposed. Method of
approximate solution that results in the substantial decrease of the data transmitted is brought forward |
| format |
Article |
| author |
Dadyka, D.I. Anisimov, I.O. |
| author_facet |
Dadyka, D.I. Anisimov, I.O. |
| author_sort |
Dadyka, D.I. |
| title |
Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method |
| title_short |
Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method |
| title_full |
Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method |
| title_fullStr |
Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method |
| title_full_unstemmed |
Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method |
| title_sort |
direct parallel poisson solver with the multiply grid spacing support for plasma simulation via pic method |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Basic plasma |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/115322 |
| citation_txt |
Direct parallel Poisson solver with the multiply grid spacing support for plasma simulation via PIC method / D.I. Dadyka, I.O. Anisimov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 81-83. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT dadykadi directparallelpoissonsolverwiththemultiplygridspacingsupportforplasmasimulationviapicmethod AT anisimovio directparallelpoissonsolverwiththemultiplygridspacingsupportforplasmasimulationviapicmethod |
| first_indexed |
2025-07-08T08:35:29Z |
| last_indexed |
2025-07-08T08:35:29Z |
| _version_ |
1837067120744071168 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2016. №6(106)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 81-83. 81
DIRECT PARALLEL POISSON SOLVER WITH THE MULTIPLY GRID
SPACING SUPPORT FOR PLASMA SIMULATION VIA PIC METHOD
D.I. Dadyka, I.O. Anisimov
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
E-mail: d.dadyka@gmail.com
Problem of the plasma simulation area decomposition for Poisson equation solution via PIC method using
parallel computing is treated. Modification of the existing method moving to reduction of the computational
complexity is discussed. The approach to use different grid spacing for subdomains is proposed. Method of
approximate solution that results in the substantial decrease of the data transmitted is brought forward.
PACS: 02.60.Pn, 52.65.Rr, 52.80.Tn
INTRODUCTION
Computer simulation via PIC method
demonstrates the high accuracy of the plasma
behavior description [1]. For the wide class of
problems (e.g., for beam-plasma discharge – see [2])
the characteristic length of the system is of the order
of 10
3
Debye radii and the simulation time is of the
order of 10
3
Langmuir periods. But PIC method needs
the large RAM capacity and high computing
performance. Consequently such problems need the
clusters which are often based on single program,
multiple data (SPMD) with the distributed memory
architecture [3]. Algorithms used for these clusters
must satisfy such important conditions as minimal
volume of data transferred between the mesh nodes
and uniform nodes' loading.
The integration of the field equations (Poisson
equation in electrostatic case) remains a difficult task,
because significant data non-locality is present.
Besides, there is a problem of dynamic balancing for
the inhomogeneous plasma case (for the beam-plasma
discharge, for example) that is not considered in the
existing literature. For inhomogeneous plasma the
Debye length can vary in space, but for correct
simulation we need to use the computation grid
spacing less than Debye length [1]. On the other hand,
small grid spacing usage is not optimal for regions with
large Debye length. Consequently it is necessary to
decompose the problem into subdomains with different
spacing. In this paper the domain decomposition
method which supports this requirement is proposed.
Besides, it is shown how to obtain maximal
computation productivity and to reduce the transmitted
data number by using approximate solving.
1. EXISTING DOMAIN DECOMPOSITION
METHOD AND ITS EXTENSION FOR
NON-UNIFORM GRIDS
Poisson equation solving method based on
decomposition of the main problem into
subdomains with zero Dirichlet boundary
conditions and calculating of the screen charges at
the subdomains' interfaces was proposed in [5]. For
subdomains' joining the author suggests to add the
Laplace equation solution with the boundary
potentials determined by interfaces' screen charges
to the Poisson equation solution for subdomains.
Recursive domain joining method is proposed.
But the above mentioned method is not valid for
subdomains with different spacing. Besides, this method
requires the multiply Laplace equation solving (4 times in
2D case).
The following formula is used for computation of the
screen charges ρscr,i on the vertical interface:
, , , ,
, 2 2
2 0left i right i left i right i
scr i
x xh h
, (1)
where Φleft,i and Φright,i are pre-boundary potentials of left
and right subdomains, respectively, and i =0...Ly.
This formula can’t be used if subdomains have different
spacing along the interface, so arrays Φleft,i1 and Φright,i2
have different sizes i1 =0...Ly1, i2 =0...Ly2.
We propose to use the spectral representation of the
preboundary potentials and screen charges, so
, ,
, 2
SP SP
left k right kSP
scr k
xh
, (2)
where k =0...max(Ly1, Ly2). This formula is independent
on the grids' step and it is correct for any Φleft,k and Φright,k
size. The missing high frequency components of
spectrum for coarse grid are vanished. For computing of
the interface potentials we use eigenvalues of the problem
with height max(Ly1, Ly2) and width Lx1+Lx2+1. It will be
shown that calculation of the spectral representations can
be obtained directly without additional 1D fast Fourier
transforms (FFTs).
Formula (2) isn’t correct in the case of subdomains
with different spacing in the direction perpendicular to
the interface, because hx is not uniform. In this case it is
necessary to modify the interface potentials' computation
algorithm. The following approach is proposed: the
joining problem with non-uniform grid spacing can be
considered as a sum of two joining problems with the
uniform spacing assuming that one of the subdomains is
vanished. So one can compute two interface charges'
distribution:
,
1, 2
1
SP
left kSP
scr k
xh
, 2,
2, 2
2
SP
right kSP
scr k
xh
. (3)
Then for each screen charges' layer ρscr1,i and ρscr2,i we
can compute potentials via formulas from [5]. Sum of the
computed potentials gives the desired interface potential.
Note that subdomains grid spacing can’t be arbitrary;
the length of combined domain must be divisible by hx1
and hx2.
82 ISSN 1562-6016. ВАНТ. 2016. №6(106)
2. MODIFICATION OF THE POISSON
EQUATION SOLVING
As it was shown in [5], joining of domains
requires two actions: solving of Poisson equation with
zero Dirichlet boundary conditions for finding the
pre-boundary potentials and solving the Laplace
equation with interfaces' potentials. The sum of these
solutions gives the correct solution for subdomain
potential. We can improve this performance by
combining Poisson and Laplace solving. Let us
consider the Poisson solving for 2D Fourier
expansion method. Using this method we need to
compute 2D spectrum of the input charges' density
11
, ,
0 0
1 1 1 1
sin sin
1 1
yx
LL
SP
k l i j
i j x y
k i l j
L L
(4)
by FFT (DST-I for this case), to divide the spectrum
by eigenvalues
1
, 2 2
,
1 1
sin sin ,
4 1 1
SP
k lSP
k l
x y
k l
L L
(5)
and then to compute the potential distribution from
the spectrum by inverse FFT
11
, ,
0 0
1 1 1 1
sin sin
1 1
yx
LL
SP
i j k l
k l x y
k i l j
L L
(6)
(the normalizing coefficient is missed). But one can
compute the pre-boundary potentials without full 2D
FFT by computing only one sum of (6) for i=0, i=Lx–
1 and j=0, j=Ly–1. For adding the Laplace equation
solution to the Poisson equation solution it is possible
to add the Laplace boundary conditions to the pre-
boundary charges of the Poisson equation [6] or to
add the spectrum of this pre-boundary charges to the
Poisson equation spectrum by using (4) for i=0,
i=Lx–1 and j=0, j=Ly–1. So only two full 2D FFTs
and O(LyLx) additional steps can be used instead of
four FFTs. It is also important that we propose to use
the spectral representation of the interface potentials.
Consequently we don’t need any additional 1D FFTs
for finding the pre-boundary and interfaces'
potentials, we can use the spectra directly.
3. OPTIMIZATION OF COMPUTATIONS
FOR THE DECOMPOSITION IN TWO
DIRECTIONS
As it was shown above, we don’t need to solve
Poisson equation and Laplace equation separately.
But in the case of decomposition in two directions we
need to compute the pre-boundary potentials when
the perpendicular interfaces' potentials are found. We
can use the Laplace equation solving with
O(LxLylog(Lx)) complexity [5]. But the better way can
be proposed. Consider computation of the bottom
pre-boundary subdomain potentials Φbottom,l when left
interface potentials Φleft,k are found. By adding
interface potentials as the pre-boundary charges of the
Poisson equation and expanding 1D charges'
spectrum into 2D Poisson spectrum we can obtain the
following formulas:
1
,
,
0 ,
1 1
sin sin
1 1
y SPL
left nSP
bottom l
n k n y x
n l
EV L L
, (7)
2 2
1 1
, 4 sin sin
1 1x y
k l
EV k n
L L
, (8)
where l=0, …, Lx–1, k=0, ..., Ly–1.
So, we can compute Φbottom,l by O(LyLx) additional steps.
4. APPROXIMATE INTERFACE POTENTIAL
FINDING TO REDUCE AMOUNT OF THE
TRANSMITTED DATA
When decomposition in two directions is used the
interface and pre-boundaries points are distributed into
several nodes. There are no effective methods for the
parallel FFT computation using the distributed memory
systems, therefore it is necessary to send all the pre-
boundary points into one main node for interface
potentials' computing and then to send the obtained
potentials to each node (Figure). The amount of data
transmitted to the main node may be too large. Besides,
each node can use the different grid spacing, so we need
to compute FFT for the non-uniform grid. It’s
inconvenient to use formulas from [5] in this case.
Another way can be proposed. Finding of the interface
potentials scr by the Fourier method can be replaced by
sum of solutions of 1D screened Poisson equations:
1
0
yL
scr l
l
, (9)
02 2
1 1
sin
1
l l l scr
y
i l
L
, (10)
2 2
2
14
sin
2 1
l
y y
l
h L
, (11)
where l=0,…,Ly is the number of spatial harmonic, i0 is
the interface position. This result can be obtained via
expanding of the Poisson solution in the single series (see
[6], p. 194).
Data transmission for layers' joining
For high spatial harmonics with l>>1 parameter l
depends weakly on l, so we can use the piecewise
approximation for (11) and reduce the number of
screened equations in the set (10).
In practice, we need only 6 pieces for approximation
of the harmonics' solution in the range n=Lx/8, …, Lx–1
ISSN 1562-6016. ВАНТ. 2016. №6(106) 83
with the computer double type precision (10
–15
). For
solving of each screened Poisson equation as the
tridiagonal problem we need to send only two values
between nodes. We use the bisection algorithm [7] for
minimizing the successive data exchange up to the
value O(log(N)), where N is the number of nodes in
the layer. In the low frequency range n=0, ..., Lx/8 we
use FFT method, so the full data transmitted number
is C'=2(Lx/8+6) (for the same number of
approximation terms) instead of C=2Lx. Note that if
we do not perform the high frequency filtration for
input ρscr, the approximate solution (9) with the
reduced number of terms will contain incorrect low
frequency components. Since we know the piecewise
approximation for αl, we can find the low frequency
dispersion of solution (9) and subtract it from FFT
low frequency solution dispersion.
For recursive domain only joining of the low
frequency components is used. So full algorithm of
the method contains the following steps (for
horizontal layers):
1. Finding of the interfaces' screen charges by
sending, e.g., the bottom layer pre-boundary potential
to the top layer. The node pre-boundary potentials are
sent only to the nodes with the common borders.
2. For each interface the high frequency solution
of the potentials for main problem is found.
3. For each interface the low frequency spectrum
of charges is computed.
4. The low frequency solution for the interface
potentials is found using the recursive domain joining
method.
5. The low frequency solutions obtained at the
step 4 are added to the high frequency solutions
obtained at the step 2.
CONCLUSIONS
1. Existing packages for PIC simulation do not
contain effective methods of parallel Poisson
equation solving.
2. Inhomogeneous plasma simulation needs the
dividing of the simulation area into subdomains with
different grid spacing.
3. Domain decomposition method for subdomains
with different spacing is proposed.
4. The scheme of calculating for minimizing the
number of computations is proposed. The proposed
calculation needs the number of steps close to the method
of expansion into the double Fourier series.
5. The transmitted data minimization method using
approximate solving is proposed.
REFERENCES
1. C.K. Birdsall, A.B. Langdon / Plasma Physics via
Computer Simulation. McGraw: “Hill book company”,
1985.
2. D.I. Dadyka, I.O. Anisimov. 2D simulation of the
initial stage of the beam-plasma discharge // Problems of
Atomic Science and Technology. Series “Plasma
Physics” (21). 2015, № 1 (95), p. 149-151.
3. D.A. Patterson, J.L. Hennessy. Computer Organization
and Design, Fourth Edition: The Hardware/Software
Interface. Morgan Kaufmann. 2011, p. 635-645.
4. F. Wolfheimer, E. Gjonaj, T. Weiland. Parallel
particle-in-cell (PIC) codes //Proceedings of International
Computational Accelerator Physics Conference,
Chamonix, France. 2006, p. 290-295.
5. N.V. Snytnikov. A parallel algorithm for solving 2D
Poisson’s equation in the context of no stationary
problems // Computational Methods and Programming.
2015, v. 16, p. 39-51.
6. A.A. Samarsky, E.S. Nikolaev. Methods for solving
grid equations. M.: “Nauka”, “Fizmatlit”. 1978 (in
Russian).
7. A.V. Terekhov. Parallel Dichotomy Algorithm for
Solving Tridiagonal System of Linear Equation with
Multiple Right-Hand Sides // Parallel Computing. 2010,
v. 36, p. 423-438.
Article received 05.10.2016
ПРЯМОЙ ПАРАЛЛЕЛЬНЫЙ МЕТОД РЕШЕНИЯ УРАВНЕНИЯ ПУАССОНА
С ИСПОЛЬЗОВАНИЕМ РАЗЛИЧНОГО ШАГА СЕТОК ДЛЯ МОДЕЛИРОВАНИЯ МЕТОДОМ
КРУПНЫХ ЧАСТИЦ В ЯЧЕЙКАХ
Д.И. Дадыка, И.А. Анисимов
Рассмотрена проблема декомпозиции области моделирования для уравнения Пуассона методом крупных
частиц в ячейках. Предложены пути минимизации вычислительных затрат. Продемонстрирована
возможность использования различного шага сеток для подобластей. Предложен метод значительного
снижения объёма передаваемых между вычислительными узлами данных.
ПРЯМИЙ ПАРАЛЕЛЬНИЙ МЕТОД РОЗВ'ЯЗАННЯ РІВНЯННЯ ПУАССОНА
З ВИКОРИСТАННЯМ РІЗНОГО КРОКУ СІТОК ДЛЯ МОДЕЛЮВАННЯ МЕТОДОМ КРУПНИХ
ЧАСТИНОК У КОМІРКАХ
Д.І. Дадика, І.О. Анісімов
Розглянуто проблему декомпозиції області моделювання для рівняння Пуассона методом крупних
частинок у комірках. Запропоновано шляхи мінімізації обчислювальних витрат. Продемонстровано
можливість використання різного кроку сіток для підобластей. Запропоновано метод значного зниження
об'єму даних, що передаються між обчислювальними вузлами.
|