41 research outputs found
A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov-Poisson Equation
Numerical solutions to the Vlasov-Poisson system of equations have important
applications to both plasma physics and cosmology. In this paper, we present a
new Particle-in-Cell (PIC) method for solving this system that is 4th-order
accurate in both space and time. Our method is a high-order extension of one
presented previously [B. Wang, G. Miller, and P. Colella, SIAM J. Sci. Comput.,
33 (2011), pp. 3509--3537]. It treats all of the stages of the standard PIC
update - charge deposition, force interpolation, the field solve, and the
particle push - with 4th-order accuracy, and includes a 6th-order accurate
phase-space remapping step for controlling particle noise. We demonstrate the
convergence of our method on a series of one- and two- dimensional
electrostatic plasma test problems, comparing its accuracy to that of a
2nd-order method. As expected, the 4th-order method can achieve comparable
accuracy to the 2nd-order method with many fewer resolution elements.Comment: 18 pages, 10 figures, submitted to SIS
A "metric" semi-Lagrangian Vlasov-Poisson solver
We propose a new semi-Lagrangian Vlasov-Poisson solver. It employs elements
of metric to follow locally the flow and its deformation, allowing one to find
quickly and accurately the initial phase-space position of any test
particle , by expanding at second order the geometry of the motion in the
vicinity of the closest element. It is thus possible to reconstruct accurately
the phase-space distribution function at any time and position by
proper interpolation of initial conditions, following Liouville theorem. When
distorsion of the elements of metric becomes too large, it is necessary to
create new initial conditions along with isotropic elements and repeat the
procedure again until next resampling. To speed up the process, interpolation
of the phase-space distribution is performed at second order during the
transport phase, while third order splines are used at the moments of
remapping. We also show how to compute accurately the region of influence of
each element of metric with the proper percolation scheme. The algorithm is
tested here in the framework of one-dimensional gravitational dynamics but is
implemented in such a way that it can be extended easily to four or
six-dimensional phase-space. It can also be trivially generalised to plasmas.Comment: 32 pages, 14 figures, accepted for publication in Journal of Plasma
Physics, Special issue: The Vlasov equation, from space to laboratory plasma
The Convergence of Particle-in-Cell Schemes for Cosmological Dark Matter Simulations
Particle methods are a ubiquitous tool for solving the Vlasov-Poisson
equation in comoving coordinates, which is used to model the gravitational
evolution of dark matter in an expanding universe. However, these methods are
known to produce poor results on idealized test problems, particularly at late
times, after the particle trajectories have crossed. To investigate this, we
have performed a series of one- and two-dimensional "Zel'dovich Pancake"
calculations using the popular Particle-in-Cell (PIC) method. We find that PIC
can indeed converge on these problems provided the following modifications are
made. The first modification is to regularize the singular initial distribution
function by introducing a small but finite artificial velocity dispersion. This
process is analogous to artificial viscosity in compressible gas dynamics, and,
as with artificial viscosity, the amount of regularization can be tailored so
that its effect outside of a well-defined region - in this case, the
high-density caustics - is small. The second modification is the introduction
of a particle remapping procedure that periodically re-expresses the dark
matter distribution function using a new set of particles. We describe a
remapping algorithm that is third-order accurate and adaptive in phase space.
This procedure prevents the accumulation of numerical errors in integrating the
particle trajectories from growing large enough to significantly degrade the
solution. Once both of these changes are made, PIC converges at second order on
the Zel'dovich Pancake problem, even at late times, after many caustics have
formed. Furthermore, the resulting scheme does not suffer from the unphysical,
small-scale "clumping" phenomenon known to occur on the Pancake problem when
the perturbation wave vector is not aligned with one of the Cartesian
coordinate axes.Comment: 29 pages, 29 figures. Accepted for publication in ApJ. The revised
version includes a discussion of energy conservation in the remapping
procedure, as well as some interpretive differences in the Conclusions made
in response to the referee report. Results themselves are unchange
A cloudy Vlasov solution
We propose to integrate the Vlasov-Poisson equations giving the evolution of
a dynamical system in phase-space using a continuous set of local basis
functions. In practice, the method decomposes the density in phase-space into
small smooth units having compact support. We call these small units ``clouds''
and choose them to be Gaussians of elliptical support. Fortunately, the
evolution of these clouds in the local potential has an analytical solution,
that can be used to evolve the whole system during a significant fraction of
dynamical time. In the process, the clouds, initially round, change shape and
get elongated. At some point, the system needs to be remapped on round clouds
once again. This remapping can be performed optimally using a small number of
Lucy iterations. The remapped solution can be evolved again with the cloud
method, and the process can be iterated a large number of times without showing
significant diffusion. Our numerical experiments show that it is possible to
follow the 2 dimensional phase space distribution during a large number of
dynamical times with excellent accuracy. The main limitation to this accuracy
is the finite size of the clouds, which results in coarse graining the
structures smaller than the clouds and induces small aliasing effects at these
scales. However, it is shown in this paper that this method is consistent with
an adaptive refinement algorithm which allows one to track the evolution of the
finer structure in phase space. It is also shown that the generalization of the
cloud method to the 4 dimensional and the 6 dimensional phase space is quite
natural.Comment: 46 pages, 25 figures, submitted to MNRA
A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov-Poisson equation
This paper presents an optimized and scalable semi-Lagrangian solver for the
Vlasov-Poisson system in six-dimensional phase space. Grid-based solvers of the
Vlasov equation are known to give accurate results. At the same time, these
solvers are challenged by the curse of dimensionality resulting in very high
memory requirements, and moreover, requiring highly efficient parallelization
schemes. In this paper, we consider the 6d Vlasov-Poisson problem discretized
by a split-step semi-Lagrangian scheme, using successive 1d interpolations on
1d stripes of the 6d domain. Two parallelization paradigms are compared, a
remapping scheme and a classical domain decomposition approach applied to the
full 6d problem. From numerical experiments, the latter approach is found to be
superior in the massively parallel case in various respects. We address the
challenge of artificial time step restrictions due to the decomposition of the
domain by introducing a blocked one-sided communication scheme for the purely
electrostatic case and a rotating mesh for the case with a constant magnetic
field. In addition, we propose a pipelining scheme that enables to hide the
costs for the halo communication between neighbor processes efficiently behind
useful computation. Parallel scalability on up to 65k processes is demonstrated
for benchmark problems on a supercomputer