Meshfree Plasma Simulation with a Parallel Tree
Code
Numerical simulation of hot, ionized matter poses
a perennial challenge to the theoretical plasma physicist
because of the virtually unlimited degrees of freedom,
extreme nonlinear behaviour and vast range of length
– and timescales characteristic of both natural
and manmade plasmas. Traditionally, the intractability
of firstprinciples simulation is overcome by first
simplifying the problem in phase space; replacing
individual particle trajectories by a smooth velocity
distribution and then solving a VlasovBoltzmanntype
equation. By rigorous application of kinetic theory,
many problems can be further reduced to the magnetohydrodynamics
picture – the plasma equivalent of the NavierStokes
equations.
Whether kinetic or fluid, nearly all plasma modelling
over the past four decades has relied on a spatial
mesh to mediate the interplay between plasma particles
and their associated electric and magnetic fields.
While these models have proved highly successful,
the presence of a grid ultimately places restrictions
on the spatial resolution or geometry which can be
considered – especially in three dimensions.
Recently a new meshfree plasma simulation paradigm
has been developed which overcomes some of these limitations.
Inspired by the Nbody tree algorithms designed to
speed up gravitational problems in astrophysics, this
approach reverts to first principles by computing
forces on individual particles directly, following
their trajectories in a Lagrangian, “molecular
dynamics” fashion [1].
Figure 1: Meshfree kinetic simulation of proton acceleration
by a high intensity, shortpulse laser
At ZAM we have applied this technique to study particle
transport in Petawatt laser interactions with solid
targets – see Figure 1. In this case the laser
(blue discs) is modelled as a simple momentum and
heat source, drilling into the target and accelerating
a substantial fraction of the plasma electrons to
energies of several MeV in the process. In effect,
the laser induces a multiMegaamp current directed
into the target, a feat which is only sustainable
if a return current can be supplied by the cold background
charge. If the target resistivity is high, an imbalance
will result, setting up a DC electric field in the
range of 1012 Vm1. This field hinders the hot electrons
from passing through the target (orange cloud), but
on the other hand leads to enhanced acceleration of
ions from the front side of the target (arrows). Such
laserbased energetic ion sources have many promising
applications in areas such as isotope production,
tumor therapy and advanced fusion schemes [2].
Figure 2: Scaling of the parallel tree code on Jump
and BlueGene/L for spheres with various numbers of
charges
A typical investigation is set up with a total of
6 million electrons and ions placed in a “foil”
with dimensions 12x12x5 µm3. The simulation
of this interaction process for a 100 fs laser pulse
would consume 5000 hours on a single Power4 CPU, but
this reduces to around 100 wallclock hours when run
on 96 processors (3 frames) of the IBM supercomputer
“Jump” at the Research Centre Jülich.
Further preliminary benchmark tests with several million
charges (125 million) demonstrate that this code
scales up to at least 256 CPUs on Jump and 1024 CPUs
on the new BlueGene/L architecture – Figure
2. Although slower than particleincell codes (their
meshbased equivalents) parallel tree codes offer
completely new possibilities in plasma simulation,
particularly where collisions are important (here
they are included automatically); or for modelling
complex geometries; or for masslimited systems in
which artificial boundaries would severely compromise
the simulation´s validity (for example atomic
clusters). The generic nature of this algorithm, combined
with good parallel scalability, means that it can
be easily adapted to other systems dominated by longrange
interactions – currently one of the research
priorities at ZAM [3].
References
[1] S. Pfalzner, P. Gibbon
Phys. Rev. E 57, 4698 (1998)
[2] P. Gibbon
Short Pulse Laser Interactions with Matter,
Imperial College Press, London (2005)
[3] For further information,
see:
www.fzjuelich.de/zam/cams
• Paul Gibbon
Central Institute for Applied Mathematics (ZAM),
Research Centre Jülich
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