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Discrete Dirac Mechanics and Geometric Numerical Integration Methods for Plasma Physics

Periodic Reporting for period 2 - DDM-GNI (Discrete Dirac Mechanics and Geometric Numerical Integration Methods for Plasma Physics)

Reporting period: 2017-04-01 to 2018-03-31

In recent years it has been realised that the preservation of certain structures and properties of mathematical equations is of utmost importance when bringing them in a form understandable by computers. Ignoring such properties often leads to incorrect results or numerical instabilities in the corresponding simulations. In fluid dynamics as well as fusion and astro plasma physics this is especially true as the mathematical equations are highly complex and direct comparison with the experiments is very difficult. Still, the application of structure-preserving numerical algorithms to plasma physics has not gained much attention and todays computer codes rarely use such methods. Consequently, these codes are not able to reproduce important physical effects observed in the experiments, hindering progress in research. This project is devoted to the development of a flexible and general framework for the derivation of geometric numerical integrators, referred to as discrete Dirac mechanics, which will be applicable to many problems from fluid dynamics and plasma physics.
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
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 ( 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.
The new Hamilton-Pontryagin-Galerkin integrators unify many known variational integrators in one common framework. In addition, they allow for the derivation of novel long-time stable integrators, especially for complicated problems like degenerate Lagrangian systems or Hamiltonian systems with Dirac constraints, for which general geometric integrators have not been available so far. The availability of this new framework will allow for the development of geometric integrators for many problems from fluid dynamics and plasma physics, for which such integrators could not be obtained in the past.

The Poisson and metriplectic integrators developed for the Vlasov-Maxwell system, the Landau collision operator and magnetohydrodynamics provide discrete representations of these systems that retain more features of the continuous equations than any other discretisation obtained before. Moreover, the discrete framework developed allows for the systematic construction of new integrators, based on different discretisation techniques, eliminating the guess work that used to be an elementary part of the construction of structure-preserving numerical methods in the past.

The proliferation of the newly developed algorithms will improve the quality of the output and the predictive capabilities of simulation codes in various research areas of fusion and astro plasma physics as well as fluid dynamics. Moreover, they will allow for simulations in regimes that are not accessible with traditional algorithms, facilitating the study of problems that cannot be simulated with state-of-the-art computer codes. Thus the deployment of new simulation tools based on the algorithms developed within the DDM-GNI project will foster the understanding of important physical phenomena like turbulent transport in fusion plasmas or the solar wind, thereby potentially allowing for the construction of smaller and thus more economic fusion reactors and the accurate solar weather prediction after eruptions on the sun.