Advanced Particle Simulation for Computational Cosmology and
Beam Physics: Beam Physics
Particle accelerators have helped enable some of the
most remarkable discoveries of the 20th century. They have also led to
substantial advances in applied science and technology, many of which
greatly benefit society. Accelerator-based systems have now been
proposed to address problems of international importance related to
energy and the environment. Given the importance of particle
accelerators, it is imperative that the most advanced high performance
computing tools be brought to bear on their design, optimization,
technology development, and operation. The SciDAC accelerator modeling
project is a national research and development effort whose primary
objective is to establish a comprehensive terascale simulation
environment needed to solve the most challenging problems in 21st
century accelerator science and technology.
The SciDAC accelerator effort is a national-scale
project, targeted mainly to machine design and optimization
applications. In the present project, we are mainly interested in the
development of next-generation modeling capabilities; this work
dovetails nicely with similar issues in cosmological simulations.
The major effort will be to develop methods for
intrabeam collisions, beam-beam interactions, beam stability, and
development of multi-grid adaptive-mesh refinement (AMR) solvers for
the Poisson equation.
Computation of intrabeam collisions is important for
several reasons. Aside from fundamental issues such as relaxation
mechanisms in beam dynamics, collisions are responsible for emittance
growth in circular machines, possible generation of beam halo in
linacs, and provide a useful mechanism for beam cooling. We have
already developed LANGEVIN3D, a particle-based code that
self-consistently solves the Fokker-Planck equation for soft
(multiple, small angle) collisions in beams. The capabilities of this
code will be extended to multiple particle species and error control
will be rigorously tested in inhomogeneous situations.
The development of AMR methods in particle simulation
is useful not only to improve resolution but also to better integrate
with multi-resolution electromagnetics solvers. Multi-grid solvers
have a proven record in solving the Poisson equation and these methods
are well-matched to AMR strategies. We will develop parallel solvers
for the Poisson equation and investigate error control issues such as
particle splitting, particle-grid interaction, and AMR mesh generation
in addition to performance optimization.
In many cases of physical interest, such as intense
beams, one needs to take into account the mean force field of all
other particles on the particle of interest (the Vlasov-Poisson
equation) as well as account for the soft collisions. The inclusion of
a Fokker-Planck collision term on the right hand side of the Vlasov
equation gives rise to the Landau equation. The Landau equation is a
partial differential equation with self-consistently determined
systematic force terms as well as external fields, if present, and
self-consistent friction and diffusion coefficients arising from the
Fokker-Planck treatment of collisions. Determination of all the
self-consistent contributions requires the computation of convolution
integrals in either real or velocity space.
A successful approach to modeling the Vlasov-Poisson
equation is the popular PIC technique where simulation particles are
used to indirectly represent the phase space distribution function and
the Poisson equation is solved on a spatial grid. The advantages of
the PIC method include its relative conceptual simplicity, high
performance resulting from fast Poisson solvers, relatively low memory
cost for the grid (O(L^k) where k is the number of spatial
dimensions), and insensitivity to the generation of small-scale
structure in the distribution function. Moreover, PIC simulations for
accelerator applications have been implemented efficiently on parallel
computing platforms. Fokker-Planck collisions can be included in the
PIC method via the addition of friction and (multiplicative)
stochastic forces in the equations of motion for the simulation
particles: This is the Langevin approach to incorporating soft
collisions. It should be kept in mind that numerical collisions are
present in any PIC simulation of the type just described. Thus, it is
appropriate to include the physical collisions only when the numerical
collisions are strongly suppressed in the original Vlasov-Poisson
simulation. This condition can be met in some situations of interest.
The main difficulty in carrying out the Langevin PIC
program is the fact that the self-consistent friction and diffusion
coefficients themselves depend on the velocity, thus, in principle,
for every simulation particle one needs to carry out two convolution
integrals in velocity space followed by appropriate derivatives, also
in velocity space. Given the PIC point of view, one would wish to
introduce a velocity grid associated with each spatial grid cell,
carry out the convolutions on the velocity grid and then use
interpolation to determine the appropriate friction and diffusion
coefficients for the simulation particles belonging to that particular
spatial grid cell. These tasks have been viewed as being much too
difficult to actually carry out: either the Spitzer approximation has
been employed or an isotropic velocity distribution has been assumed
for the scattering particles.
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However, we have shown that on modern parallel
machines these problems can be overcome (in large part) and the fully
self-consistent friction and diffusion coefficients obtained
numerically for any distribution. In short, the key points are that
the velocity grids need not be very large (we found 32^3 to be
sufficient), one may associate a single velocity grid not with a
single spatial grid cell but with some number of them (a form of
coarse-graining), the number of particles associated with each spatial
super-cell is large enough to guarantee low sampling noise in velocity
space, and finally, the convolution and interpolation strategies
already implemented for the spatial part of the Vlasov-Poisson
equation may be directly extended to velocity space.
As a test case, we have applied our numerical method
to compute the friction and diffusion coefficients for a Maxwellian
velocity distribution, the results of which are shown in the figures
on the right. An important point that is clearly demonstrated is the
modest number of particles needed per spatial super-cell to reach
convergence of the computed quantities. The top figure shows the
diagonal and off-diagonal diffusion coefficients for a Maxwellian
distribution as a function of velocity. While the Spitzer
approximation would result in a straight line, here the expected
fall-off in the velocity is clearly seen and excellent results are
obtained even for a small number of sampled particles (there is
essentially no difference between 3000 and 1.25 million particles).
The asymptotic fall-off in the self-consistent dynamical friction
coefficient F_d/v (v is the velocity) as 1/v^3 at large v is seen
nicely in the bottom figure (log-log scale). The red curve is the
numerical result, while the expected 1/v^3 asymptotic slope is shown
by the green curve; the agreement is very good.
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| Salman Habib / LANL / revised November 03 |
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