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

• Open Science, Open Access, Open Source

• Julia @ Home Tutorial I: Language Overview

• Julia @ Home Tutorial II: Advanced Julia

• Julia @ Home Assignments

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-Ampère 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-Poincaré 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