scholarly journals Algorithms of motion in the particle-in-cell method

2021 ◽  
pp. 281-293
Author(s):  
Е.С. Воропаева ◽  
К.В. Вшивков ◽  
Л.В. Вшивкова ◽  
Г.И. Дудникова ◽  
А.А. Ефимова
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Charged Particle ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Charged Particles ◽  
Two Dimensional ◽  
Time Step ◽  
Particle In Cell ◽  
Particles In Electromagnetic Fields

В настоящей работе представлен новый метод решения уравнений движения заряженных частиц в электромагнитных полях и проведено его сравнение с различными известными модификациями метода Бориса. Созданные двумерный и трехмерный алгоритмы основаны на использовании точного решения дифференциального уравнения для скорости заряженной частицы на шаге по времени. Сравнительный анализ метода Бориса и его модификаций проводился как по точности методов, так и по времени их работы. Новая модификация метода Бориса позволяет точнее вычислять траекторию и скорость заряженной частицы без значительного увеличения сложности расчетов. Показано, что при выборе модификации метода Бориса для решения задачи в первую очередь следует обращать внимание на точность решения, так как более простая и быстрая схема может не дать выигрыша по времени. The article proposes a new method for solving the equations of motion of charged particles in electromagnetic fields and compares this method with various known modifications of the Boris method. The created two-dimensional and three-dimensional algorithms are based on the use of an exact solution of the differential equation for the velocity of a charged particle at a time step. A comparative analysis of the Boris method and its modifications was carried out both in terms of the accuracy of the methods and the time of their operation. A new modification of the Boris method allows more accurate calculations of the trajectory and velocity of a charged particle without a significant increase in the complexity of calculations. It is shown that, when choosing a modification of the Boris method to solve a problem, one should pay attention first of all to the accuracy of the solution, since a simpler and faster scheme may not give a gain in time.

Motion of a charged particle in superposed Heliotron and biconical cusp fields

1969 ◽  
Vol 3 (2) ◽  
pp. 255-267 ◽  
Author(s):  
M. P. Srivastava ◽  
P. K. Bhat
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Magnetic Moment ◽  
Particle Trajectory ◽  
Three Dimensional ◽  
Equation Of Motion ◽  
Neutral Point ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Hybrid Type

We have studied the behaviour of a charged particle in an axially symmetric magnetic field having a neutral point, so as to find a possibility of confining a charged particle in a thermonuclear device. In order to study the motion we have reduced a three-dimensional motion to a two-dimensional one by introducing a fictitious potential. Following Schmidt we have classified the motion, as an ‘off-axis motion’ and ‘encircling motion’ depending on the behaviour of this potential. We see that the particle performs a hybrid type of motion in the negative z-axis, i.e. at some instant it is in ‘off-axis motion’ while at another instant it is in ‘encircling motion’. We have also solved the equation of motion numerically and the graphs of the particle trajectory verify our analysis. We find that in most of the cases the particle is contained. The magnetic moment is found to be moderately adiabatic.

Two-dimensional motion of a parabolically confined charged particle in a perpendicular magnetic field

Open Physics ◽  
2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Orion Ciftja
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Equations Of Motion ◽  
Complex Variables ◽  
Perpendicular Magnetic Field ◽  
Two Dimensional ◽  
Cyclotron Motion ◽  
Lissajous Figures ◽  
Long Time ◽  
Concentric Circles

AbstractThe classical two-dimensional motion of a parabolically confined charged particle in presence of a perpendicular magnetic is studied. The resulting equations of motion are solved exactly by using a mathematical method which is based on the introduction of complex variables. The two-dimensional motion of a parabolically charged particle in a perpendicular magnetic field is strikingly different from either the two-dimensional cyclotron motion, or the oscillator motion. It is found that the trajectory of a parabolically confined charged particle in a perpendicular magnetic field is closed only for particular values of cyclotron and parabolic confining frequencies that satisfy a given commensurability condition. In these cases, the closed paths of the particle resemble Lissajous figures, though significant differences with them do exist. When such commensurability condition is not satisfied, path of particle is open and motion is no longer periodic. In this case, after a sufficiently long time has elapsed, the open paths of the particle fill a whole annulus, a region lying between two concentric circles of different radii.

Implementation of 2D Domain Decomposition in the UCAN Gyrokinetic Particle-in-Cell Code and Resulting Performance of UCAN2

2016 ◽  
Vol 19 (1) ◽  
pp. 205-225 ◽  
Author(s):  
Jean-Noel G. Leboeuf ◽  
Viktor K. Decyk ◽  
David E. Newman ◽  
Raul Sanchez
Keyword(s):  
Domain Decomposition ◽  
Parallel Implementation ◽  
Three Dimensional ◽  
Massively Parallel ◽  
Problem Size ◽  
Time Step ◽  
Particle In Cell ◽  
Minor Radius ◽  
Efficient Charge ◽  
Cartesian Geometry

AbstractThe massively parallel, nonlinear, three-dimensional (3D), toroidal, electrostatic, gyrokinetic, particle-in-cell (PIC), Cartesian geometry UCAN code, with particle ions and adiabatic electrons, has been successfully exercised to identify non-diffusive transport characteristics in present day tokamak discharges. The limitation in applying UCAN to larger scale discharges is the 1D domain decomposition in the toroidal (or z-) direction for massively parallel implementation using MPI which has restricted the calculations to a few hundred ion Larmor radii or gyroradii per plasma minor radius. To exceed these sizes, we have implemented 2D domain decomposition in UCAN with the addition of the y-direction to the processor mix. This has been facilitated by use of relevant components in the P2LIB library of field and particle management routines developed for UCLA's UPIC Framework of conventional PIC codes. The gyro-averaging specific to gyrokinetic codes is simplified by the use of replicated arrays for efficient charge accumulation and force deposition. The 2D domain-decomposed UCAN2 code reproduces the original 1D domain nonlinear results within round-off. Benchmarks of UCAN2 on the Cray XC30 Edison at NERSC demonstrate ideal scaling when problem size is increased along with processor number up to the largest power of 2 available, namely 131,072 processors. These particle weak scaling benchmarks also indicate that the 1 nanosecond per particle per time step and 1 TFlops barriers are easily broken by UCAN2 with 1 billion particles or more and 2000 or more processors.

scholarly journals TACHYON EFFECTS ON THE TWO-DIMENSIONAL BLACK HOLE GEOMETRY

1995 ◽  
Vol 10 (18) ◽  
pp. 1277-1286 ◽  
Author(s):  
G.A. DIAMANDIS ◽  
B.C. GEORGALAS ◽  
E. PAPANTONOPOULOS
Keyword(s):  
Exact Solution ◽  
Black Hole ◽  
Black Hole Solution ◽  
Equations Of Motion ◽  
Tree Level ◽  
Two Dimensional ◽  
Full System ◽  
Dimensional Black Hole ◽  
Hole Geometry ◽  
Hole Structure

We study solutions of the tree level string effective action in the presence of the tachyon mode. In the case of static fields we find numerically that the full system has a black hole solution with the tachyon regular at the horizon. We also find a nonstatic exact solution of the equations of motion having a black hole structure with a past singularity.

scholarly journals Scalar-Torsion Theories in Four Dimensional Static Spacetimes

2021 ◽  
Vol 32 (2) ◽  
pp. 12-15
Author(s):  
Mulyanto . ◽  
Fiki Taufik Akbar ◽  
Bobby Eka Gunara
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solution ◽  
Master Equation ◽  
Equations Of Motion ◽  
Scalar Potential ◽  
Torsion Theory ◽  
Linear Ordinary Differential Equation ◽  
Four Dimensions ◽  
Torsion Theories

In this paper, we consider a class of static spacetimes scalar-torsion theories in four dimensioanal static spacetimes with the scalar potential turned on. We discover that the 2-dimensional submanifold must admit constant triplet structures, one of which is the torsion scalar. This indicates that these equations of motion can be reduced to a single highly non-linear ordinary differential equation known as the master equation. Then, we show that there are no exact solution of the scalar-torsion theory in four dimensions considering the Sinh-Gordon potential.

scholarly journals EXPERIENCE OF OBTAINING THE PLAN OF EXPERIMENTS, WITH USING MODELS OF INTERACTION OF CHARGED PARTICLES

2017 ◽  
Vol 2 ◽  
pp. 47
Author(s):  
Normunds Kante ◽  
Juris Lavedels ◽  
N. Kriščuks
Keyword(s):  
Charged Particle ◽  
Particle Density ◽  
Dimensional Space ◽  
Charged Particles ◽  
Multidimensional Space ◽  
Two Dimensional ◽  
Infinite Space ◽  
Surface Models ◽  
Charged Particle Density

In this article a method of obtaining an experiment plan in a fragment of multidimensional space is analyzed and improved. The method is based on an assumption that particles will distribute evenly in an infinite space with constant charged particle density. To obtain the experiment plan, the infinite multidimensional space is replaced with a hypercube whose surface models influence of the surrounding infinite space. Software is developed and practical results in obtaining experiment plan in two-dimensional space are acquired. Two-dimensional space allows developing of a methodology and algorithm for obtaining experiment plan while providing a simple visualization of the solution. Acquired results in two-dimensional space give an opportunity to create methods for obtaining experiment plan in a hypercube of multidimensional space.

A New Method for the Solution of a Differential Equation with Two-Point Boundary Conditions Applied to the Compressible Boundary Layer on a Yawed Infinite Wing

1956 ◽  
Vol 60 (552) ◽  
pp. 808-809
Author(s):  
L. F. Crabtree ◽  
E.R. Woollett
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Exact Solution ◽  
Stagnation Point ◽  
Close Agreement ◽  
Equations Of Motion ◽  
New Method ◽  
Compressible Boundary Layer ◽  
Point Boundary ◽  
Direct Solution

The compressible laminar boundary layer on a yawed infinite wing is considered in Ref. 1, where it is shown that the problem may be solved by a direct solution of the linearised equations of motion under certain assumptions. As an example of this procedure the boundary layer near a stagnation point was calculated. Tinkler has published solutions of the exact equations for the general Falkner-Skan case (Ref. 1) obtained on the M.I.T. differential analyser for several values of the parameter involved. It has been found that the numerical results of Ref. 1 were in error and the corrected results obtained by a new method are tabulated below. Tinkler's exact solution of the stagnation point flow for ω = 0·10 is also given for comparison, and it will be seen that there is close agreement

Time-Domain Simulations of Radiation and Diffraction Forces

2010 ◽  
Vol 54 (02) ◽  
pp. 79-94 ◽  
Author(s):  
Xinshu Zhang ◽  
Piotr Bandyk ◽  
Robert F. Beck
Keyword(s):  
Free Surface ◽  
Time Domain ◽  
Three Dimensional ◽  
The Body ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Forward Speed ◽  
Time Step ◽  
Surface Boundary Conditions ◽  
Strip Theory

Large-amplitude, time-domain, wave-body interactions are studied in this paper for problems with forward speed. Both two-dimensional strip theory and three-dimensional computation methods are shown and compared by a number of numerical simulations. In the present approach, an exact body boundary condition and linearized free surface boundary conditions are used. By distributing desingularized sources above the calm water surface and using constant-strength flat panels on the exact body surface, the boundary integral equations are solved numerically at each time step. The strip theory method implements Radial Basis Functions to approximate the longitudinal derivatives of the velocity potential on the body. Once the fluid velocities on the free surface are computed, the free surface elevation and potential are updated by integrating the free surface boundary conditions. After each time step, the body surface and free surface are regrided due to the instantaneous changing wetted body geometry. Extensive results are presented to validate the efficiency of the present methods. These results include the added mass and damping computations for a Wigley III hull and an S-175 hull with forward speed using both two-dimensional and three-dimensional approaches. Exciting forces acting on a Wigley III hull due to regular head seas are obtained and compared using both the fully three-dimensional method and the two-dimensional strip theory. All the computational results are compared with experiments or other numerical solutions.

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