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

TL;DR

This paper introduces degenerate variational integrators that provide stable, structure-preserving numerical solutions for magnetic field line flow and guiding center trajectories, overcoming limitations of previous multistep methods.

Contribution

The work develops one-step variational integrators for non-canonical Hamiltonian systems by preserving degeneracy, enhancing stability and qualitative accuracy over prior multistep approaches.

Findings

Demonstrated superior stability over existing variational integrators.

Achieved better qualitative behavior than non-conservative algorithms.

Preserved non-canonical symplectic structure in numerical simulations.

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.