222 research outputs found
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
WENO schemes applied to the quasi-relativistic Vlasov--Maxwell model for laser-plasma interaction
In this paper we focus on WENO-based methods for the simulation of the 1D
Quasi-Relativistic Vlasov--Maxwell (QRVM) model used to describe how a laser
wave interacts with and heats a plasma by penetrating into it. We propose
several non-oscillatory methods based on either Runge--Kutta (explicit) or
Time-Splitting (implicit) time discretizations. We then show preliminary
numerical experiments
A semi-Lagrangian Vlasov solver in tensor train format
In this article, we derive a semi-Lagrangian scheme for the solution of the
Vlasov equation represented as a low-parametric tensor. Grid-based methods for
the Vlasov equation have been shown to give accurate results but their use has
mostly been limited to simulations in two dimensional phase space due to
extensive memory requirements in higher dimensions. Compression of the solution
via high-order singular value decomposition can help in reducing the storage
requirements and the tensor train (TT) format provides efficient basic linear
algebra routines for low-rank representations of tensors. In this paper, we
develop interpolation formulas for a semi-Lagrangian solver in TT format. In
order to efficiently implement the method, we propose a compression of the
matrix representing the interpolation step and an efficient implementation of
the Hadamard product. We show numerical simulations for standard test cases in
two, four and six dimensional phase space. Depending on the test case, the
memory requirements reduce by a factor in four and a factor
in six dimensions compared to the full-grid method
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
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