5,716 research outputs found
Numerical simulations of the Fourier transformed Vlasov-Maxwell system in higher dimensions --- Theory and applications
We present a review of recent developments of simulations of the
Vlasov-Maxwell system of equations using a Fourier transform method in velocity
space. In this method, the distribution functions for electrons and ions are
Fourier transformed in velocity space, and the resulting set of equations are
solved numerically. In the original Vlasov equation, phase mixing may lead to
an oscillatory behavior and sharp gradients of the distribution function in
velocity space, which is problematic in simulations where it can lead to
unphysical electric fields and instabilities and to the recurrence effect where
parts of the initial condition recur in the simulation. The particle
distribution function is in general smoother in the Fourier transformed
velocity space, which is desirable for the numerical approximations. By
designing outflow boundary conditions in the Fourier transformed velocity
space, the highest oscillating terms are allowed to propagate out through the
boundary and are removed from the calculations, thereby strongly reducing the
numerical recurrence effect. The outflow boundary conditions in higher
dimensions including electromagnetic effects are discussed. The Fourier
transform method is also suitable to solve the Fourier transformed Wigner
equation, which is the quantum mechanical analogue of the Vlasov equation for
classical particles.Comment: 41 pages, 19 figures. To be published in Transport Theory and
Statistical Physics. Proceedings of the VLASOVIA 2009 Workshop, CIRM, Luminy,
Marseilles, France, 31 August - 4 September 200
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 "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
Kinetic Vlasov Simulations of collisionless magnetic Reconnection
A fully kinetic Vlasov simulation of the Geospace Environment Modeling (GEM)
Magnetic Reconnection Challenge is presented. Good agreement is found with
previous kinetic simulations using particle in cell (PIC) codes, confirming
both the PIC and the Vlasov code. In the latter the complete distribution
functions () are discretised on a numerical grid in phase space.
In contrast to PIC simulations, the Vlasov code does not suffer from numerical
noise and allows a more detailed investigation of the distribution functions.
The role of the different contributions of Ohm's law are compared by
calculating each of the terms from the moments of the . The important role
of the off--diagonal elements of the electron pressure tensor could be
confirmed. The inductive electric field at the X--Line is found to be dominated
by the non--gyrotropic electron pressure, while the bulk electron inertia is of
minor importance. Detailed analysis of the electron distribution function
within the diffusion region reveals the kinetic origin of the non--gyrotropic
terms
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
Vlasov simulation in multiple spatial dimensions
A long-standing challenge encountered in modeling plasma dynamics is
achieving practical Vlasov equation simulation in multiple spatial dimensions
over large length and time scales. While direct multi-dimension Vlasov
simulation methods using adaptive mesh methods [J. W. Banks et al., Physics of
Plasmas 18, no. 5 (2011): 052102; B. I. Cohen et al., November 10, 2010,
http://meetings.aps.org/link/BAPS.2010.DPP.NP9.142] have recently shown
promising results, in this paper we present an alternative, the Vlasov Multi
Dimensional (VMD) model, that is specifically designed to take advantage of
solution properties in regimes when plasma waves are confined to a narrow cone,
as may be the case for stimulated Raman scatter in large optic f# laser beams.
Perpendicular grid spacing large compared to a Debye length is then possible
without instability, enabling an order 10 decrease in required computational
resources compared to standard particle in cell (PIC) methods in 2D, with
another reduction of that order in 3D. Further advantage compared to PIC
methods accrues in regimes where particle noise is an issue. VMD and PIC
results in a 2D model of localized Langmuir waves are in qualitative agreement
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