April 4-6

Abstract Details

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Author: Joshua W Burby
Requested Type: Poster
Submitted: 2022-03-04 10:18:33

Co-authors: E. Hirvijoki, M. Leok

Contact Info:
Los Alamos National Laboratory
P.O. Box 1663
Los Alamos, New Mexico   87547

Abstract Text:
Non-dissipative (i.e. Hamiltonian) dynamical systems freeze flux in phase space, just as highly-conductive plasma flows freeze magnetic flux. A time integrator for a non-dissipative system is symplectic when it freezes flux exactly. Symplectic integration is routine in canonical coordinates, where the flux tensor takes the simplest possible form. Much less is understood about symplectic integration in the general non-canonical case, which occurs more frequently in practice. I will present a general approach to structure-preserving integration of noncanonical Hamiltonian systems on exact symplectic manifolds. First, the original non-canonical Hamiltonian system is embedded in a larger (essentially) canonical system as a slow manifold. Then a canonical symplectic integrator for the larger system is identified that has approximately the same slow manifold. Provided initial conditions are selected near the slow manifold, the integrator provides a good approximation of the original system. There would be a problem with this approach if the discrete-time slow manifold happened to have any normal instabilities; such instabilities would carry discrete trajectories away from the slow manifold, and the good approximation properties would break down. I will explain how this potential problem is avoided using a newly-developed theory of nearly-periodic maps. By constraining the large system's integrator to be a non-resonant nearly-periodic map, existence of a discrete-time adiabatic invariant is guaranteed. Long-time normal stability of the slow manifold then follows from a Lyapunov-type argument.

fits in computer simulation of plasmas category