31 research outputs found
Conservative and non-conservative methods based on hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations
We introduce a WENO reconstruction based on Hermite interpolation both for
semi-Lagrangian and finite difference methods. This WENO reconstruction
technique allows to control spurious oscillations. We develop third and fifth
order methods and apply them to non-conservative semi-Lagrangian schemes and
conservative finite difference methods. Our numerical results will be compared
to the usual semi-Lagrangian method with cubic spline reconstruction and the
classical fifth order WENO finite difference scheme. These reconstructions are
observed to be less dissipative than the usual weighted essentially non-
oscillatory procedure. We apply these methods to transport equations in the
context of plasma physics and the numerical simulation of turbulence phenomena
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Geometric Numerical Integration (hybrid meeting)
The topics of the workshop
included interactions between geometric numerical integration and numerical partial differential equations;
geometric aspects of stochastic differential equations;
interaction with optimisation and machine learning;
new applications of geometric integration in physics;
problems of discrete geometry, integrability, and algebraic aspects
Variational Integrators in Plasma Physics
Variational integrators are a special kind of geometric discretisation
methods applicable to any system of differential equations that obeys a
Lagrangian formulation. In this thesis, variational integrators are developed
for several important models of plasma physics: guiding centre dynamics
(particle dynamics), the Vlasov-Poisson system (kinetic theory), and ideal
magnetohydrodynamics (plasma fluid theory). Special attention is given to
physical conservation laws like conservation of energy and momentum.
Most systems in plasma physics do not possess a Lagrangian formulation to
which the variational integrator methodology is directly applicable. Therefore
the theory is extended towards nonvariational differential equations by linking
it to Ibragimov's theory of integrating factors and adjoint equations. It
allows us to find a Lagrangian for all ordinary and partial differential
equations and systems thereof. Consequently, the applicability of variational
integrators is extended to a much larger family of systems than envisaged in
the original theory. This approach allows for the application of Noether's
theorem to analyse the conservation properties of the system, both at the
continuous and the discrete level.
In numerical examples, the conservation properties of the derived schemes are
analysed. In case of guiding centre dynamics, momentum in the toroidal
direction of a tokamak is preserved exactly. The particle energy exhibits an
error, but the absolute value of this error stays constant during the entire
simulation. Therefore numerical dissipation is absent. In case of the kinetic
theory, the total number of particles, total linear momentum and total energy
are preserved exactly, i.e., up to machine accuracy. In case of
magnetohydrodynamics, the total energy, cross helicity and the divergence of
the magnetic field are preserved up to machine precision.Comment: PhD Thesis, 222 page
Turbulence-driven ion beams in space plasmas
The description of the local turbulent energy transfer and the high-resolution ion distributions measured by the Magnetospheric Multiscale mission together provide a formidable tool to explore the cross-scale connection between the fluid-scale energy cascade and plasma processes at subion scales. When the small-scale energy transfer is dominated by Alfv´enic, correlated velocity, and magnetic field fluctuations, beams of accelerated particles are more likely observed. Both space observations and numerical simulations suggest the nonlinear wave-particle interaction as one possible mechanism for the energy dissipation in space plasmas