Abstract:

Problems in plasma physics are rich in geometric structure. This is especially true when considering kinetic models (ones which describe the evolution of a probability density function for the charged particles, typically through hyperbolic PDEs) as they possess infinite-dimensional Hamiltonian / Poisson structure. Exploiting this structure has led to the development of advanced numerical methods for solving kinetic problems very accurately.

Issues arise, however, when one is interested in adding collisions to such models, as collisions are dissipative and do not fit into the Hamiltonian structure of the problem. This led to the development of the metriplectic formulation in the eighties, which is a geometric way of describing Hamiltonian systems with dissipation. This formulation can also be utilised to construct accurate numerical discretisations when modelling collisions, which is a much more recent topic of study.

In this talk, I will introduce the relevant plasma physics models and their geometric structure. I will then present the metriplectic formulation, and detail how it can be used to describe Hamiltonian systems with dissipation. Finally, I will describe how the metriplectic structure of collision operators can be used to construct discretisations in a way that exactly preserves the quantities of interest.