Most of the work performed in the first period of the project was dedicated to the derivation of a new Hamilton-Pontryagin-Galerkin integrator framework and the development of important theoretical structures from Dirac mechanics within this framework, namely the discrete variational structure, the discrete symplectic structure and the discrete Noether theorem. This new framework has then been used to develop new families of discontinuous Galerkin variational integrators for degenerate Lagrangian systems and for Hamiltonian systems subject to Dirac constraints. The new framework was applied to several test problems, including point vortices and guiding centre dynamics, in order to obtain novel long-time stable integrators for these systems. These new integrators were then implemented in a new Julia library, which is publicly and freely available at
https://github.com/DDMGNI/GeometricIntegrators.jl(s’ouvre dans une nouvelle fenêtre).
In the second period, another new family of integrators, so called projected variational integrators for degenerate Lagrangian systems, was developed, applied to the guiding centre problem, and implemented in the GeometricIntegrators.jl library.
Further, work was performed on the evaluation of possible discretisation strategies for systems on Lie groups, as they are common in field theories from fluid dynamics and plasma physics. A promising discretisation strategy for the variational formulation of systems on Lie groups based on Discontinuous Galerkin methods has been identified. First theoretical results for simple systems in one dimension show the viability of this approach. The extension towards more complicated systems in two and three dimensions is currently underway in a follow-up project.
A novel framework for the particle-in-cell discretisation of the Hamiltonian formulation of systems from plasma physics and fluid dynamics has been developed and applied to the Vlasov-Maxwell system and magnetohydrodynamics. The resulting integrators have been implemented in the SeLaLib library (
http://selalib.gforge.inria.fr/(s’ouvre dans une nouvelle fenêtre)). Both, theoretical analysis and numerical experiments, show the unprecedented conservation properties of these new integrators, in particular with respect to the Poisson structure, conservation of energy and charge as well as important Casimir invariants like Gauss' law and the divergence of the magnetic field.
In order to facilitate the treatment of dissipative effects, a discretisation strategy for so-called metriplectic brackets based on Finite Element Methods was devised. The resulting metriplectic integrators have been applied to the Vlasov-Maxwell-Landau system, leading to a discretisation of the Landau collision operator that preserves all important thermodynamical properties, namely energy conservation, entropy production and an H-theorem.
All major results have been published in peer-reviewed journals and presented at workshops and conferences. The transfer of the newly developed algorithms to application codes is underway. In particular, some of the novel guiding centre integrators have already been implemented in the ORB5 and HAGGIS codes, which are used for physics studies of fast particle dynamics and turbulent transport.
