Symplectic integrators offer many benefits for numerically approximating solutions to Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two important 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. New algorithms were recently developed using the variational integration formalism; however, those integrators were found to admit parasitic mode instabilities due to their multistep character. 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 the resultant Euler-Lagrange equations are systems of first-order ordinary differential equations. We show that retaining the same degree of degeneracy when constructing discrete Lagrangians yields one-step variational integrators preserving a non-canonical symplectic structure. Numerical examples demonstrate the benefits of the new algorithms, including superior stability relative to the existing variational integrators for these systems and superior qualitative behavior relative to non-conservative algorithms.
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Research Article|
May 04 2018
Degenerate variational integrators for magnetic field line flow and guiding center trajectories
C. L. Ellison;
C. L. Ellison
a)
1
Lawrence Livermore National Laboratory
, Livermore, California 94550, USA
2
Princeton Plasma Physics Laboratory
, Princeton, New Jersey 08543, USA
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J. M. Finn;
J. M. Finn
b)
3
Los Alamos National Laboratory
, Los Alamos, New Mexico 87545, USA
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J. W. Burby;
J. W. Burby
4
Courant Institute of Mathematical Sciences
, New York, New York 10012, USA
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M. Kraus
;
M. Kraus
5
Max-Planck-Institut für Plasmaphysik
, Garching, Germany
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H. Qin;
H. Qin
2
Princeton Plasma Physics Laboratory
, Princeton, New Jersey 08543, USA
6
Department of Modern Physics, University of Science and Technology of China
, Hefei, Anhui 230026, China
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W. M. Tang
W. M. Tang
2
Princeton Plasma Physics Laboratory
, Princeton, New Jersey 08543, USA
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a)
Electronic mail: ellison6@llnl.gov.
b)
Present address: Tibbar Plasma Technologies, 274 DP Rd., Los Alamos, NM 87544, USA.
Phys. Plasmas 25, 052502 (2018)
Article history
Received:
January 12 2018
Accepted:
April 13 2018
Citation
C. L. Ellison, J. M. Finn, J. W. Burby, M. Kraus, H. Qin, W. M. Tang; Degenerate variational integrators for magnetic field line flow and guiding center trajectories. Phys. Plasmas 1 May 2018; 25 (5): 052502. https://doi.org/10.1063/1.5022277
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