# Geometric Methods for Physics of Magnetised Plasmas

Many models in plasma physics have been shown to exhibit a Hamiltonian structure and can be derived from an action principle. This is true for the equations of motion of a particle in a given electromagnetic field, which is a finite dimensional Hamiltonian system, for which symplectic numerical integrators have been derived with a large success and a well developed theory. Many other, more complex models for plasma physics, disregarding dissipative effects, also fit into an infinite dimensional non canonical hamiltonian geometric structure. This geometric structure provides the basis for the conservation of some fundamental physics invariants like energy, momentum and some Casimir invariants like Gauss' law and div B=0. Therefore preserving it in asymptotic models, like for example the Gyrokinetic model for strongly magnetised plasmas, or numerical approximations can be very helpful. New numerical methods, like mimetic Finite Differences and Finite Element Exterior Calculus based on the discretisation of objects coming from differential geometry allow in a natural way to preserve the geometric structure of the continuous equations. Schemes derived on these concepts allow on the one hand to rederive very good well-known schemes that have previously been found in ad hoc way, like the Yee scheme for Maxwell's equations or charge conserving Particle In Cell methods. These concepts will be introduced and applied to some classical models from plasma physics: Maxwell's equations, a cold plasma model and the Vlasov-Maxwell equations.## Outline of the lectures

- Variational principles: Finite dimensional and infinite dimensional (field theories)
- Basic notions of differential geometry and their discretisation: Manifolds, differential forms, exterior derivative, interior product, ...
- Mimetic Finite Differences and their application to plasma physics models
- Finite Element Exterior Calculus and its application to plasma physics models
- Geometric derivation of modern gyrokinetic models and related invariants

## Exercise classes

The lectures will be complemented with an exercise class where the notions developed in the lecture will be applied to concrete examples and codes for the discrete models will be written in Python.## News

- No lecture on Monday 1 February
- Questions and answer session on Monday 8 February
- Final exam (oral): Friday 12 February 2021

## Literature

- V.I. Arnold, Mathematical methods of classical mechanics (Vol. 60). Springer Science & Business Media (2013).
- D. Arnold, R. Falk, R. Winther. Finite Element Exterior Calculus, from Hodge theory to numerical stability, Bull. AMS, vol 47 (2), pp. 281--354 (2010).
- E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Springer 2006.
- M. Kraus, K. Kormann, P. J. Morrison, E. Sonnendrücker. GEMPIC: Geometric electromagnetic particle-in-cell methods. Journal of Plasma Physics, 83(4) (2017).
- J. Kreeft, A. Palha, M. Gerritsma: Mimetic framework on curvilinear quadrilaterals of arbitrary order. arXiv preprint arXiv:1111.4304 (2011).
- K. Lipnikov, G. Manzini, M. Shashkov. Mimetic finite difference method. Journal of Computational Physics, 257, 1163-1227 (2014).
- P. J.Morrison. Structure and structure-preserving algorithms for plasma physics. Physics of Plasmas, 24(5), 055502 (2017).

## Lecturers

- Lectures: Prof. Dr. Eric Sonnendrücker. Monday 10:15 - 11:45 online via Zoom
- Exercise classes: Irene Garnelo Abellanas. Monday 12:00 - 12:45 online via Zoom