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