649 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
Afterlive: A performant code for Vlasov-Hybrid simulations
A parallelized implementation of the Vlasov-Hybrid method [Nunn, 1993] is
presented. This method is a hybrid between a gridded Eulerian description and
Lagrangian meta-particles. Unlike the Particle-in-Cell method [Dawson, 1983]
which simply adds up the contribution of meta-particles, this method does a
reconstruction of the distribution function in every time step for each
species. This interpolation method combines meta-particles with different
weights in such a way that particles with large weight do not drown out
particles that represent small contributions to the phase space density. These
core properties allow the use of a much larger range of macro factors and can
thus represent a much larger dynamic range in phase space density.
The reconstructed phase space density is used to calculate momenta of the
distribution function such as the charge density . The charge density
is also used as input into a spectral solver that calculates the
self-consistent electrostatic field which is used to update the particles for
the next time-step.
Afterlive (A Fourier-based Tool in the Electrostatic limit for the Rapid
Low-noise Integration of the Vlasov Equation) is fully parallelized using MPI
and writes output using parallel HDF5. The input to the simulation is read from
a JSON description that sets the initial particle distributions as well as
domain size and discretization constraints. The implementation presented here
is intentionally limited to one spatial dimension and resolves one or three
dimensions in velocity space. Additional spatial dimensions can be added in a
straight forward way, but make runs computationally even more costly.Comment: Accepted for publication in Computer Physics Communication
Direct Integration of the Collisionless Boltzmann Equation in Six-dimensional Phase Space: Self-gravitating Systems
We present a scheme for numerical simulations of collisionless
self-gravitating systems which directly integrates the Vlasov--Poisson
equations in six-dimensional phase space. By the results from a suite of
large-scale numerical simulations, we demonstrate that the present scheme can
simulate collisionless self-gravitating systems properly. The integration
scheme is based on the positive flux conservation method recently developed in
plasma physics. We test the accuracy of our code by performing several test
calculations including the stability of King spheres, the gravitational
instability and the Landau damping. We show that the mass and the energy are
accurately conserved for all the test cases we study. The results are in good
agreement with linear theory predictions and/or analytic solutions. The
distribution function keeps the property of positivity and remains
non-oscillatory. The largest simulations are run on 64^6 grids. The computation
speed scales well with the number of processors, and thus our code performs
efficiently on massively parallel supercomputers.Comment: 35 pages, 19 figures. Submitted to the Astrophysical Journa
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 PDD method for solving linear, nonlinear, and fractional PDEs problems
We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel. For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves). Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators. The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.info:eu-repo/semantics/acceptedVersio
A discontinuous Galerkin method for the Vlasov-Poisson system
A discontinuous Galerkin method for approximating the Vlasov-Poisson system
of equations describing the time evolution of a collisionless plasma is
proposed. The method is mass conservative and, in the case that piecewise
constant functions are used as a basis, the method preserves the positivity of
the electron distribution function and weakly enforces continuity of the
electric field through mesh interfaces and boundary conditions. The performance
of the method is investigated by computing several examples and error estimates
associated system's approximation are stated. In particular, computed results
are benchmarked against established theoretical results for linear advection
and the phenomenon of linear Landau damping for both the Maxwell and Lorentz
distributions. Moreover, two nonlinear problems are considered: nonlinear
Landau damping and a version of the two-stream instability are computed. For
the latter, fine scale details of the resulting long-time BGK-like state are
presented. Conservation laws are examined and various comparisons to theory are
made. The results obtained demonstrate that the discontinuous Galerkin method
is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86
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