2018-03-21
10:15 at HCP F 43.4

On the structure-preserving discretisation of Poisson and metriplectic brackets

Michael Kraus

Max Planck Institute for Plasma Physics, Garching, Germany

Many conservative systems in physics feature a Hamiltonian structure, that is their dynamics can be described in terms of a bilinear operator, called Poisson bracket, and an energy functional, called the Hamiltonian. In order to include certain non-ideal effects, this formulation can be extended by a metric bracket and an entropy functional that is dissipated, leading to the so-called metriplectic framework. Both formulations, the purely Hamiltonian as well as the metriplectic, feature a rich geometric structure including various families of conservation laws. In this talk, we present novel approaches to the structure-preserving discretisation of systems that are either Hamiltonian or metriplectic, exemplified by systems from plasma physics.
First, we present a novel framework for Finite Element Particle-in-Cell methods of the Vlasov–Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, namely anti-symmetry, the Jacobi identity, and conservation of Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, Hamiltonian and operator splitting methods are used for integration in time. The resulting methods are gauge-invariant, feature exact charge conservation and show excellent long-time energy and momentum behavior.
Second, we present a novel framework for Galerkin methods of the nonlinear Landau and other collision operators. Using a Finite Element discretization for the velocity space, we transform the infinite-dimensional metriplectic system into a finite-dimensional, time-continuous metriplectic system. Temporal discretisation is accomplished using discrete gradients. The resulting integrators feature exact conservation of energy, momentum, and mass, as well as the production of entropy.