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E. Metral et al.
4.7 Numerical Modelling
G. Rumolo
Collective effects can be studied analytically, either through the perturbation
formalism applied to the Vlasov equation or by means of few (typically two
or three) particles models with the basic ingredients such as to reproduce the
essential features of the phenomenon under study. Both ways are usually based
on simplified approaches, in which some assumptions are necessary to make the
models analytically solvable and lead to limited sets of equations relatively easy to
interpret and handy to use.
The analytically solvable two or three particles models can be refined further to
more realistic models, in which more than just few particles are assumed to represent
the full particle beam. However, since the number of coupled differential equations
to be solved grows proportionally with the number of particles used in the model,
the resulting set of equations will rapidly become unmanageable as we increase the
number of macroparticles (which are used to approximate the beam with a reduced
number of particles), unless it is fed into a numerical simulation to be run on a
computer. By using computers, the number of macroparticles necessary to model
a beam can be pushed up to several millions, which is very useful to study the
details of all possible internal oscillation modes of a bunch (or train of bunches),
and also incoherent effects like emittance growth. Although ideally we would like
to develop simulation programs that take into account the highest possible number
of effects, in practice the existing codes narrow down their models to one or few
effects, whose consequences in the beam dynamics are interesting to single out.
For example, to study the effects of electron clouds, the beam will be made to
interact with a given electron cloud at some locations around the accelerator ring,
but in general other possible interactions with impedances, or the concurrent effects
of space charge, beam–beam, Intra Beam Scattering, will be neglected. Although
the study of two or more effects simultaneously is technically possible in most
cases, at the present state of art of the simulations, it is generally preferred to
limit the study of such interplays, because the combined models are difficult to
control and tend to break down. Pushing further on this line, not only different
effects can be decoupled in simulations, but also different regimes can be studied
separately in the beam dynamics. For instance, for some problems only a partial
description of the beam will be sufficient, so that transverse problems can be treated
separately from longitudinal problems as well as single-bunch/multi-turn effects can
be studied ignoring that these bunches are parts of long trains. In some cases, singlebunch/single- or multi-turn effects can also be modelled to generate driving terms
to be used in reduced studies extending over longer time scales.
To perform a simulation, we will therefore have to define our beam as an
ensemble of macroparticles, identified through arrays containing the phase space
coordinates of each macro-particle (2–6-dimensional). This beam is first initialized
and then transported across selected points of the accelerator ring using the
E. Metral et al.
4.7 Numerical Modelling
G. Rumolo
Collective effects can be studied analytically, either through the perturbation
formalism applied to the Vlasov equation or by means of few (typically two
or three) particles models with the basic ingredients such as to reproduce the
essential features of the phenomenon under study. Both ways are usually based
on simplified approaches, in which some assumptions are necessary to make the
models analytically solvable and lead to limited sets of equations relatively easy to
interpret and handy to use.
The analytically solvable two or three particles models can be refined further to
more realistic models, in which more than just few particles are assumed to represent
the full particle beam. However, since the number of coupled differential equations
to be solved grows proportionally with the number of particles used in the model,
the resulting set of equations will rapidly become unmanageable as we increase the
number of macroparticles (which are used to approximate the beam with a reduced
number of particles), unless it is fed into a numerical simulation to be run on a
computer. By using computers, the number of macroparticles necessary to model
a beam can be pushed up to several millions, which is very useful to study the
details of all possible internal oscillation modes of a bunch (or train of bunches),
and also incoherent effects like emittance growth. Although ideally we would like
to develop simulation programs that take into account the highest possible number
of effects, in practice the existing codes narrow down their models to one or few
effects, whose consequences in the beam dynamics are interesting to single out.
For example, to study the effects of electron clouds, the beam will be made to
interact with a given electron cloud at some locations around the accelerator ring,
but in general other possible interactions with impedances, or the concurrent effects
of space charge, beam–beam, Intra Beam Scattering, will be neglected. Although
the study of two or more effects simultaneously is technically possible in most
cases, at the present state of art of the simulations, it is generally preferred to
limit the study of such interplays, because the combined models are difficult to
control and tend to break down. Pushing further on this line, not only different
effects can be decoupled in simulations, but also different regimes can be studied
separately in the beam dynamics. For instance, for some problems only a partial
description of the beam will be sufficient, so that transverse problems can be treated
separately from longitudinal problems as well as single-bunch/multi-turn effects can
be studied ignoring that these bunches are parts of long trains. In some cases, singlebunch/single- or multi-turn effects can also be modelled to generate driving terms
to be used in reduced studies extending over longer time scales.
To perform a simulation, we will therefore have to define our beam as an
ensemble of macroparticles, identified through arrays containing the phase space
coordinates of each macro-particle (2–6-dimensional). This beam is first initialized
and then transported across selected points of the accelerator ring using the