88 research outputs found
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
GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
We present a novel framework for Finite Element Particle-in-Cell methods
based on the discretization of the underlying Hamiltonian structure of the
Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains
the defining properties of a bracket, anti-symmetry and the Jacobi identity, as
well as conservation of its Casimir invariants, implying that the semi-discrete
system is still a Hamiltonian system. In order to obtain a fully discrete
Poisson integrator, the semi-discrete bracket is used in conjunction with
Hamiltonian splitting methods for integration in time. Techniques from Finite
Element Exterior Calculus ensure conservation of the divergence of the magnetic
field and Gauss' law as well as stability of the field solver. The resulting
methods are gauge invariant, feature exact charge conservation and show
excellent long-time energy and momentum behaviour. Due to the generality of our
framework, these conservation properties are guaranteed independently of a
particular choice of the Finite Element basis, as long as the corresponding
Finite Element spaces satisfy certain compatibility conditions.Comment: 57 Page
Metriplectic Integrators for the Landau Collision Operator
We present a novel framework for addressing the nonlinear Landau collision
integral in terms of finite element and other subspace projection methods. We
employ the underlying metriplectic structure of the Landau collision integral
and, using a Galerkin discretization for the velocity space, we transform the
infinite-dimensional system into a finite-dimensional, time-continuous
metriplectic system. Temporal discretization is accomplished using the concept
of discrete gradients. The conservation of energy, momentum, and particle
densities, as well as the production of entropy is demonstrated algebraically
for the fully discrete system. Due to the generality of our approach, the
conservation properties and the monotonic behavior of entropy are guaranteed
for finite element discretizations in general, independently of the mesh
configuration.Comment: 24 pages. Comments welcom
Handling the divergence constraints in Maxwell and Vlasov-Maxwell simulations
International audienceThe aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the Vlasov-Maxwell equations and the Maxwell equations with sources. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equations, for which additional boundary conditions are in some cases non trivial to find. The source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete Gauss' law that needs to be defined in accordance with the discretization of the Maxwell propagation operator
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