397 research outputs found
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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A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system
We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation
Discontinuous Galerkin methods for the Multi-dimensional Vlasov-Poisson problem
We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system
High order and energy preserving discontinuous Galerkin methods for the Vlasov-Poisson system
We present a computational study for a family of discontinuous Galerkin
methods for the one dimensional Vlasov-Poisson system that has been recently
introduced. We introduce a slight modification of the methods to allow for
feasible computations while preserving the properties of the original methods.
We study numerically the verification of the theoretical and convergence
analysis, discussing also the conservation properties of the schemes. The
methods are validated through their application to some of the benchmarks in
the simulation of plasma physics.Comment: 44 pages, 28 figure
Energy-conserving discontinuous Galerkin methods for the Vlasov-Amp\`{e}re system
In this paper, we propose energy-conserving numerical schemes for the
Vlasov-Amp\`{e}re (VA) systems. The VA system is a model used to describe the
evolution of probability density function of charged particles under self
consistent electric field in plasmas. It conserves many physical quantities,
including the total energy which is comprised of the kinetic and electric
energy. Unlike the total particle number conservation, the total energy
conservation is challenging to achieve. For simulations in longer time ranges,
negligence of this fact could cause unphysical results, such as plasma self
heating or cooling. In this paper, we develop the first Eulerian solvers that
can preserve fully discrete total energy conservation. The main components of
our solvers include explicit or implicit energy-conserving temporal
discretizations, an energy-conserving operator splitting for the VA equation
and discontinuous Galerkin finite element methods for the spatial
discretizations. We validate our schemes by rigorous derivations and benchmark
numerical examples such as Landau damping, two-stream instability and
bump-on-tail instability
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