# Research

My main interests are the geometric description of field theories from plasma physics (e.g., kinetic theories, magnetohydrodynamics) and structure preserving discretisation methods for these problems. I am also working on particle integrators (guiding centre dynamics), particle-in-cell schemes (Vlasov-Maxwell) and stochastic problems like collisions and geometric dissipation mechanisms. More recently, discontinuous Galerkin methods, reduced complexity modelling and scientific machine learning have become the focus of my research.

Many of our problems are similar to problems from fluid dynamics, with the added complication of electromagnetic fields. This means we have all the fluid waves, plus electromagnetic waves, plus the plethora of phenomena that arises when the two talk to each other. So if you think fluid dynamics is too boring, you should consider working in plasma physics.

Due to this complexity it is nontrivial to develop reliable computer codes. Standard methods often fail, either due to stability problems or because they do not capture critical features of the underlying physics. In order to obtain a more accurate discrete representation of the physical systems we use information from their geometric description. When developing new discretisation schemes we try to preserve as much of the geometric structure of the continuous equations as possible. Most often, this leads to more stable, more reliable numerical schemes than traditional methods.

## Research Software Engineering and Scientific Computing with Julia

## Ongoing Work

Structure-preserving Scientific Machine Learning

Structure-preserving Model Order Reduction and Low-rank Approximation

Geometric Discretisation of

Degenerate Lagrangian and Noncanonical Hamiltonian Systems

Vlasov-Poisson, Vlasov-AmpĂ¨re and Vlasov-Maxwell Systems

Ideal, Reduced and Full Magnetohydrodynamics

Guiding Centre Dynamics, Drift Kinetics and Gyro Kinetics

Discretisation of Action Principles, Poisson and Metriplectic Brackets using

Discontinuous Galerkin Methods

Finite Element Exterior Calculus, Mimetic Spectral Elements, B-Splines

Particle-in-Cell Methods

Discrete Dirac Mechanics and Hamilton-Pontryagin Principles

Discrete Euler-PoincarĂ© and Lie-Dirac Reduction

Discrete Noether Theorems and Conservation Laws

## Some References

Numerical Methods, Geometric Discretisation Methods, Discrete Differential Geometry

Differential Geometry, Geometric Mechanics, Multisymplectic Field Theory

Kinetic, Gyrokinetic and Gyrofluid Theory, Guiding Centre Dynamics