C. Birdsall, A. Langdon, Plasma Physics Via Computer Simulation, Institute of Physics Publishing, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....chosen. Section 3 describes the parallelization strategy adopted for the shared memory model and implemented in OpenMP. Section 4 reports the experimental results obtainde with askeletonized version of a real PIC code. 2 The plasma particle simulation application Particle simulation codes [1] seem to be the most suited tool for the investigation of turbulent plasma behaviour. Particle simulation indeed consists in evolving the phase space coordinates of a particle population in the electromagnetic elds selfconsistently computed, at each time step, in terms of the contribution yielded ....

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (McGraw-Hill, New York, 1985).

....charged ions and fast electrons forms an ionized gas called a plasma. Familiar examples of plasma include the Aurora Bore alis, neon signs, the ionosphere, and solar winds. Fusion energy is also an important application area of plasma physics research. The plasma Particle In Cell simulation model [8] integrates in time the trajectories of millions of charged particles in their self induced electro magnetic fields. Particles can be located anywhere in the spatial domain; however, the field quantities are calculated on a fixed grid. Following [60] our simulation models only the electrostatic ....

C. K. Birdsall and A. B. Langdon. Plasma Physics via Computer Simulation. The Adam Hilger Series on Plasma Physics. Adam Hilger, New York, 1991. 105 106

....where it is used to simulate the VlasovMaxwell or the Vlasov Poisson system. For convergence results for these systems we refer to [4, 5, 6, 28, 29] and in particular to [22] where the spherically symmetric Vlasov Poisson system is considered. General background on such schemes can be found in [2, 3]. For the present system particular diculties arise due to the fact that the source terms in the eld equations contribute to the elds also in a pointwise sense; as opposed to the Vlasov Poisson and Vlasov Maxwell systems they are not necessarily integrated in space(time) This lack of smoothing ....

C. K. Birdsall, A. B. Langdon. Plasma Physics via Computer Simulation. McGraw-Hill (1985).

....in each cluster fits within the processor s cache. For static irregular codes, such as sparse matrix vector multiplication, the number of clusters and their sizes are determined statically (once during runtime) similar to [11] For dynamic scientific applications, such as Particle In Cell [1] and CHARMM [2] the clusters scopes can change. Run time management used to maintain the reference clusters is similar to local memory datareallocation done for load balancing. The remainder of the paper is structured as follows. Section 2 describes the RTRC model itself and application types ....

C. K. Birdsall and A. B. Langdon. Plasma Physics via Computer Simulation. McGraw-Hill, New York, 1981.

....charged ions and fast electrons forms an ionized gas called a plasma. Familiar examples of plasma include the Aurora Borealis, neon signs, the ionosphere, and solar winds. Fusion energy is also an important application area of plasma physics research. The plasma Particle In Cell simulation model [1] integrates in time the trajectories of millions of charged particles in their self consistent electromagnetic fields. Particles can be located anywhere in the spatial domain; however, the field quantities are calculated on a fixed grid. Following [6] our simulation models only the electrostatic ....

C. K. Birdsall and A. B. Langdon. Plasma Physics via Computer Simulation. The Adam Hilger Series on Plasma Physics. Adam Hilger, New York, 1991.

....a possibly different processor, we require that the computation superstep not include operations with side effects such as output, memory allocation, etc. 4 Plasma Simulation We have implemented the lay replication scheme in the plasma simulation [9] The plasma Particle In Cell simulation model [1] integrates in time the tra jectories of millions of charged particles in their self consistent electromagnetic fields. Particles can be located anywhere in the spatial domain; however, the field quantities are calculated on a fixed grid. In the General Concurrent Particle in Cell (GCPIC) ....

C. K. Birdsall and A. B. Langdon. Plasma Physics via Computer Simulation. The Adam Hilger Series on Plasma Physics. Adam Hilger, New York, 1991.

....the equivalent C versions, yet the abstraction modeling capabilities used for scientific programming are comparably powerful. 1 Introduction Computer simulations are very useful for understanding and predicting the transport of particles and energy in fusion energy devices called tokamaks [1]. Tokamaks, which are toroidal in shape, confine the plasma with a combination of an external toroidal magnetic field and a self generated poloidal magnetic field. The plasma confinement in these devices is not well understood and is worse than desired. One of the two computer models used in the ....

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, The Adam Hilger Series on Plasma Physics, Adam Hilger, New York, 1991.

....The free electrons can transport a current; thus fusion energy is an important area of plasma physics research, but more familiar examples include the aurora borealis, neon signs, the ionosphere, and solar winds. The fundamentals of the Particle In Cell plasma simulation model are described in [1]. The model integrates in time the trajectories of large numbers of charged particles in their self consistent electrostatic (coulomb) fields. The PIC method assumes that particles do not interact with each other directly, but through the fields which they produce according to the Maxwell s ....

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, The Adam Hilger Series on Plasma Physics, Adam Hilger, New York, 1991.

....1D Electrostatic Code ES1 is a well known and documented 1D electrostatic code, written by Birdsall and Langdon, which is widely used as a teaching aid. For a more complete description of the code and a variety of simulation projects to which it can be applied, the reader is referred to reference [4]. At this point we shall present a quick outline. 2.1 Integration of the Equations of Motion The particle equations of motion are dx i dt = v i dv i dt = F i m i ES1 uses a leap frog scheme which is second order accurate in time for constant integration timestep Deltat. The discrete ....

C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill Inc., (1981).

....our conclusions. 2 THE PARALLEL CODE Since the publication of the monograph by Hockney and Eastwood in 1981 [17] a new class of particle simulation methods [2] 5] 18] 21] 22] has emerged as an alternative to particle particle (PP) 1] 14] 25] particle mesh (PM) for a review of this method see [10]) and particle particleparticle mesh (P 3 M) 13] methods. These new methods are characterized by the particles being arranged into a hierarchy of clusters, which span the full range of length scales from the minimum interparticle spacing up to the diameter of the entire system. These methods ....

C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulation (McGraw-Hill International, New York, 1985).

....of the exact transition rate S, see [24] 11] 17] for a justification of this expression and [17] 20] for a presentation of a fast algorithm to compute the collision operator. For the approximation of E i (t) we considered the classical Particle in Cell (PIC) method, see [22] [6], 7] and references therein for a presentation and [27] 9] for an error analysis. The approximations of the other macroscopic quantities, which depend on position and time, are defined at a fixed grid mesh points Xm = m Deltax by using a numerical quadrature and a regularisation function W , ....

Birdsall and Langdon, Plasma Physics via Computer Simulations, McGrawHill, New York (1985).

....simulation runs we took special care of possible non physical, i.e. numerical instabilities. Usually numerical instabilities are considered ignorable, as a rough rule of thumb, if the grid distance # does not exceed the Debye length #D = p kB T=4#n o e 2 by more than a factor of #, # ###D #cf. Birdsall and Langdon, 1991, pp. 179#. Since the main irreversible interaction is supposed to take place inside the current sheet we aim at resolving the whole sheet, by,say,twenty grid spaces, i.e. L z =10#. Then, in order to meet the numerical stability condition, the sheet half thickness L z should not exceed 30 #D . ....

....keV : 2:3 MeV and kB T e # 80 eV : 1:3 keV . In the beginning we used the real proton mass and equal mass electrons, i.e. M i = m e = M p . In these runs we had to chose kB T =5MeV in order to resolve at least three Debye length by the grid spacing as needed according to the old rule of thumb #Birdsall and Langdon, 1991#. To verify the results of this #rst series of runs we increased the resolution of the Debye length up to ten times. In order to achieve this high resolution we reduced the particle mass to the electron mass and the temperature to 40 keV . In the following section we present the results of a ....

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Birdsall, C.K., and A.B. Langdon, Plasma Physics via Computer Simulation, IOP Publishing, Plasma Physics Series #1991#.

....in general binary fluid mixtures. To simulate our system we modeled the Boltzmann collisional part using a stochastic algorithm due to Bird [15] known as Direct Simulation Monte Carlo (DSMC) while for the Vlasov part we used the particle to grid weighting method, well known in plasma physics [16]. In the DSMC method the physical space is divided into cells containing typically tens of particles. The main ingredients of this procedure are the alternation of free flow over a time interval Deltat and representative collisions among pairs of particles sharing the same cell. In the ....

....made necessary the use of two spatial grids: a somewhat coarse one for the collisions and a finer grid for the long range potential. It also imposed the use of quadratic spline interpolation for the calculation of grid quantities and a ten point difference scheme for the calculation of the forces [16]. Our results were obtained using a system with 1382400 particles in a cube with periodic boundary conditions. We also studied smaller systems to identify unavoidable finite size 3 effects. The interaction potential used was gaussian, U(x) ff Gamma 3 2 e Gammax 2 , ff 0, but there ....

C.K. Birdsall and A.B. Langdon, Plasma Physics Via Computer Simulation, (McGrawHill Book Company, New York, 1985).

....which reduces (6) to a system of N (N is the number of macroparticles) ordinary differential equations (7) 8) Here, h is the normalized contribution function. A conservative difference scheme for solving problem (1) 8) can be constructed in accordance with the method set forth in papers [12 14]. NONLINEAR DYNAMICS OF THE FLOW For the specified geometry, 1 the main control parameter that affects the behavior of the considered system is the ratio of the beam current to the ultimate vacuum current. 1 The influence of the drift chamber geometry on the VC dynamics has been studied ....

Birdsall, C.K. and Langdon, A.B., Plasma Physics via Computer Simulation, NewYork: McGraw-Hill, 1985.

....section numerical experiments that illustrate the results of Sections 2 and 3. 4.1 The linear case In a rst series of experiments, we attempt to nd a solution to the linear stationary Vlasov equation (1.1a) for a given electric potential . Numerically, we use a particle in cell method (see [10]) which enables us to compute the solution f e = f e (t; x; v) of the evolution problem: 8 : f e t v r x f e e m e r r v f e = S e (x; v) f e j = 0 ; f e (0; x; v) 0 ; 4.1) and we look for the behavior of f e (t; x; v) as t 1. Recall that this ....

Birdsall C.K., Langdon A.B., Plasma physics via computer simulation, McGraw-Hill (1985).

....were developed. The simulation of the long time evolution initiated by the instability is complicated by the presence of space charge. It is a well known fact that space charge together with the noise inherent to typical ParticleIn Cell (PIC) codes leads to artificial heating in these codes [9]. Therefore a direct noise free integration method was implemented, in addition to a usual PIC code, in order to study the long time evolution of the observables. 2 KINETIC DESCRIPTION Let 0 = 0 be the angular frequency, v 0 the velocity of the synchronous particle and 0 Delta , ....

....other side we can take advantage of the simulation noise and use the PIC code to predict the structure of the experimentally observable Schottky noise spectrum. PIC codes are well established in theoretical plasma physics, where they are used routinely to study various nonlinear kinetic phenomena [9]. In one dimensional applications one usually requires the grid spacing to resolve the Debye length ( Deltaz . D ) otherwise numerical instabilities cause substantial artificial heating [9] In an ideal plasma the phase velocity of electron plasma waves decreases for lower wave lengths. Waves ....

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C.K. Birdsall, A.B. Langdon, Plasma Physics via Computer Simulation, IOP (1991)

....known leap frog algorithm of plasma physics simulations) are widely used in the treatment of Hamiltonian systems [3 4 5] The self fields are determined by numerically solving Poisson s equation on a 3 D Cartesian grid in the beam s rest frame. Standard fast Fourier transform techniques are used [6, 7]. For an accurate representation of the 3 D fields from a bunched beam, we must use grid sizes of up to 64 x 64 x 512 nodes, as well as 10 5 10 6 macroparticles. Even with this many particles, noise effects in the fields remain an issue. 2.2 Circuit Equation for RF Cavities The RF ....

C. K. Birdsall and A. B. Langdon, Plasma Physics Via Computer Simulation, Adam Hilger, Bristol, 1991.

....and emphasize HPF features used in individual phases of the simulation code. We assess the power of HPF in representing such complicated codes. For the sake of simplicity, the most of the following discussion involves only the one dimensional PIC method; the interested reader may refer to [9, 10] for higher dimensional PIC methods. SCATTER Compute local charge densities Update charges on partition border Copy Q to QC FIELD SOLVE Inverse FFT Solve Poisson Equation FFT PARTICLE PUSH GATHER INITIALIZATION Initialize particle data Initialize grid(field) data Arrange particle and field ....

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, New York, 1985.

.... measure form of the Vlasov equation Deltat = Phi Deltat 0 ( 2:18) Convergence is proved under the assumption of a uniform convergent difference scheme Phi Deltat t and a Lipschitz continuous force F [17, 24] A simple and convergent method is the so called leap frog scheme [4], which is explicit, symmetric and second order in time, and belongs to the class of symplectic methods for hamiltonian systems [14] The simulation scheme for the Vlasov equation using the leap frog scheme for a given initial approximation N = ff N ; x; v) N ) reads (V) simulation ....

....In the case of the classical Boltzmann equation, Greengard and Reyna [15] have proved that the nonconservative Nanbu scheme always run into the zero temperature state, although that it conserves momentum and energy in the mean. In collisionless plasma simulation schemes it is well known [4] that a consistent interpolation between the particle positions and the grid of the electric field is necessary to neclect purely numerical instabilities. In the next sections modifications and extensions of the algorithms (V) and (B) are discussed which fulfill exactly the conservation equations ....

Birdsall, C.K.; Langdon, A.B.: Plasma Physics via Computer Simulation, McGraw--Hill, New York (1981)

....of the latest supercomputer. There had to be a way to preserve that investment. The POOMA FrameWork grew out of the Object Oriented Particle Simulation (OOPS) class library [Reynders et al. 1994, Reynders et al. 1995] developed at Los Alamos specifically for particle in cell (PIC) simulations [Birdsall Langdon 1985] of fusion plasmas using gyrokinetic methods [Lee 1987] Performing efficient PIC simulations is notoriously difficult on parallel architectures [Decyk 1995] PIC codes written with objects from the OOPS class libraries, however, allowed PIC simulations to move between parallel architectures with ....

C. K. Birdsall and A. B. Langdon. Plasma Physics Via Computer Simulation. McGraw-Hill, New York, 1985.

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C. Birdsall, A. Langdon, Plasma Physics Via Computer Simulation, Institute of Physics Publishing, 1991.

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Birdsall, C. K., Langdon, A. B., 1985. Plasma Physics via Computer simulations. Graw-Hill, New York.

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C.K. Birdsall and A.B. Langdon. Plasma Physics via Computer Simulations. McGraw-Hill, New York, 1981.

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C. K. Birdsall and A. Bruce Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, New York #1985#.

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C. K. Birdsall and A. B. Langdon. Plasma physics via computer simulation. Adam Hilger, Bristol, Philadelphia, 1991.

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