Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories

Abstract

Symplectic integrators offer many advantages for the numerical solution of Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two of the central Hamiltonian systems encountered in plasma physics --- the flow of magnetic field lines and the guiding center motion of magnetized charged particles --- resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates, i.e., coordinates lacking the familiar $(q,p)$ partitioning. Recent efforts made progress toward non-canonical symplectic integration of these systems by appealing to the variational integration framework; however, those integrators were multistep methods and later found to be numerically unstable due to parasitic mode instabilities. This work eliminates the multistep character and, therefore, the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem ``degenerate variational integration''. Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that their resultant Euler-Lagrange equations are systems of first-order ODEs. We show that retaining the same degree of degeneracy when constructing a discrete Lagrangian yields one-step variational integrators preserving a non-canonical symplectic structure on the original Hamiltonian phase space. The advantages of the new algorithms are demonstrated via numerical examples, demonstrating superior stability compared to existing variational integrators for these systems and superior qualitative behavior compared to non-conservative algorithms.

References in corpus (5)

• GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
• High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields
• Toroidal regularization of the guiding center Lagrangian
• The Accurate Particle Tracer Code
• Projected Variational Integrators for Degenerate Lagrangian Systems

Cited by in corpus (12)

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• Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices
• Toroidal regularization of the guiding center Lagrangian
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• Explicit Structure-Preserving Geometric Particle-in-Cell Algorithm in Curvilinear Orthogonal Coordinate Systems and Its Applications to Whole-Device 6D Kinetic Simulations of Tokamak Physics
• Slow manifold reduction for plasma science
• Field theory and structure-preserving geometric particle-in-cell algorithm for drift wave instability and turbulence
• Constrained Hamiltonian Systems and Physics-Informed Neural Networks: Hamilton-Dirac Neural Networks
• Normal stability of slow manifolds in nearly-periodic Hamiltonian systems
• Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields
• Learning of discrete models of variational PDEs from data
• Machine learning and serving of discrete field theories -- when artificial intelligence meets the discrete universe