the transport coefficients or use either the fluid or the kinetic description in different
regions of the calculation volume, but the applicability of such a model is still to be
shown.
8.2.2 Monte-Carlo Models
The mainstream approach to describing neutral particle transport in the edge plasma
is presently direct Monte-Carlo modeling [46, 47]. In this, stochastic, approach,
instead of solving the kinetic equations, Eq. (6.1), directly, one models trajectories of
the test particles, playing their interactions with the fixed background with the use of
the pseudo-random numbers generated by the computer. The source terms for
Eq. (8.1) are evaluated by calculating their intensity along the particle trajectory
and summing it over the trajectories in each grid cell. This method is versatile with
respect to the geometry of the problem, the composition of the particles considered
and the choice of different reactions they participate in by collisions with the
background particles. The latter represent, first of all, the electrons and ions of the
plasma. In non-linear models, where the neutral-neutral interactions or radiation
transport are included, the corresponding background is calculated in iterations on a
grid of trilateral cells, which covers the regular grid used for solving Eqs. (8.1) plus
the space between this grid and the material walls [42, 48–50]. The geometry can be
arbitrarily complex, including sometimes 3D objects such as downpipes [51] or
treating the full 3D configuration in a combination with a 3D plasma solver [52]. The
selection of the test particles and background species, as well as of the reactions
between them, is virtually unlimited – provided that the necessary cross-section or
rate data are available. These data are normally imported from external databases
[53] that are expandable. All this makes the Monte-Carlo approach very flexible and
convenient.
However, the Monte-Carlo approach has three principal drawbacks. First, the
numerical noise is always present in the solution because of a finite number of the
test particles traced, and this noise reduces but slowly with an increase of this
number. Therefore, Eqs. (8.1) become a set of differential equations with noisy
sources, which creates certain problems for the numerics. In particular, the highorder discretization of Eqs. (8.1) [5] becomes inefficient when a Monte-Carlo
package is used to compute the sources. Given the non-linear nature of Eq. (8.1),
even a purely random noise on the source terms there can produce bias on
the solution [54]. There are different sampling methods proposed [55], which reduce
the noise by using some information from previous iterations, either correlating the
pseudo-random number sequences between the iterations or by averaging the results
over a certain number of iterations. These methods are still under development and
are not routinely used in the 2D modeling applications, although some studies show
their efficiency, see e.g. [56].
Secondly, the calculations become slow, especially when the geometry or reaction detail need to be resolved [57]. This problem can be alleviated by the use of
8.2 Neutral Transport Models
209
regions of the calculation volume, but the applicability of such a model is still to be
shown.
8.2.2 Monte-Carlo Models
The mainstream approach to describing neutral particle transport in the edge plasma
is presently direct Monte-Carlo modeling [46, 47]. In this, stochastic, approach,
instead of solving the kinetic equations, Eq. (6.1), directly, one models trajectories of
the test particles, playing their interactions with the fixed background with the use of
the pseudo-random numbers generated by the computer. The source terms for
Eq. (8.1) are evaluated by calculating their intensity along the particle trajectory
and summing it over the trajectories in each grid cell. This method is versatile with
respect to the geometry of the problem, the composition of the particles considered
and the choice of different reactions they participate in by collisions with the
background particles. The latter represent, first of all, the electrons and ions of the
plasma. In non-linear models, where the neutral-neutral interactions or radiation
transport are included, the corresponding background is calculated in iterations on a
grid of trilateral cells, which covers the regular grid used for solving Eqs. (8.1) plus
the space between this grid and the material walls [42, 48–50]. The geometry can be
arbitrarily complex, including sometimes 3D objects such as downpipes [51] or
treating the full 3D configuration in a combination with a 3D plasma solver [52]. The
selection of the test particles and background species, as well as of the reactions
between them, is virtually unlimited – provided that the necessary cross-section or
rate data are available. These data are normally imported from external databases
[53] that are expandable. All this makes the Monte-Carlo approach very flexible and
convenient.
However, the Monte-Carlo approach has three principal drawbacks. First, the
numerical noise is always present in the solution because of a finite number of the
test particles traced, and this noise reduces but slowly with an increase of this
number. Therefore, Eqs. (8.1) become a set of differential equations with noisy
sources, which creates certain problems for the numerics. In particular, the highorder discretization of Eqs. (8.1) [5] becomes inefficient when a Monte-Carlo
package is used to compute the sources. Given the non-linear nature of Eq. (8.1),
even a purely random noise on the source terms there can produce bias on
the solution [54]. There are different sampling methods proposed [55], which reduce
the noise by using some information from previous iterations, either correlating the
pseudo-random number sequences between the iterations or by averaging the results
over a certain number of iterations. These methods are still under development and
are not routinely used in the 2D modeling applications, although some studies show
their efficiency, see e.g. [56].
Secondly, the calculations become slow, especially when the geometry or reaction detail need to be resolved [57]. This problem can be alleviated by the use of
8.2 Neutral Transport Models
209