Upcoming Event: Seminar

Discontinuous Galerkin Algorithms for a Class of Kinetic Equations in Plasma Physics

Ammar Hakim, Computational Sciences Department at Princeton Plasma Physics Laboratory

Abstract

In this talk I will present discontinuous Galerkin (DG) algorithms for several different kinetic equations that arise in plasma physics[1]. These kinetic equations evolve the distribution function of particles in 6D phase-space, with the particle motion depending on electromagnetic fields. The fields themselves are described by Maxwell equations (and its various quasi-static approximations). The fields couple to the particles via charges and currents. The DG schemes are constructed by treating the underlying dynamics as a non-canonical Hamiltonian system, using different approximation spaces for the Hamiltonian, the fields and the distribution functions. With a careful choice of numerical fluxes and approximation spaces we can construct energy conserving schemes, that are also entropy stable. When extending these schemes to arbitrary geometries, including for applications to general relativity, new challenges arise due to the extreme skewness of the momentum space when treated with standard curvilinear coordinates. To overcome this, I will show  how one can use a "tetrads-first" approach that instead uses a set of orthonormal triads/tetrads to span the tangent space of the manifold, allowing for more robust simulations with lower momentum space resolution. I will conclude with remaining challenges in constructing truly universal DG schemes for kinetic equations: robustness to positivity violations and limiters. Traditional approaches used in DG schemes for fluids do not work for kinetic equations and hence new approaches are needed. Finally, I will give some ideas on how formal proof-systems and machine-learning can be used in computational mathematics[1].

Biography

Ammar Hakim is a Principal Research Physicist and the Deputy Head of the Computational Sciences Department at Princeton Plasma Physics Laboratory. His research interests span all aspects of computational & theoretical plasma physics. He leads the Gkeyll Group that aims to develop numerical methods for plasmas at all scales[0].  [0] https://gkeyll.readthedocs.io/en/latest/ [1] See publication list at https://gkeyll.readthedocs.io/en/latest/gkeyll/pubs.html [2] J. Gorard, A. Hakim, A. (2025). “Shock with Confidence: Formal Proofs of Correctness for Hyperbolic Partial Differential Equation Solvers”, arXiv:2503.13877.