April 15-17

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Abstracts

Author: John M. Finn
Requested Type: Poster
Submitted: 2019-02-22 12:48:32

Co-authors: J. W. Burby, C. L. Ellison

Contact Info:
Tibbar Plasma Technologies, LLC
274 DP Road
Los Alamos, NM   87544
US

Abstract Text:
Degenerate variational integrators (DVIs) have recently been developed. In continuous time with n degrees of freedom, variation leads to the Hamiltonian equations, a system of 2n first order equations, where the Hessian is degenerate. Careless discretization of S can lead to a system of 2n second order equations, and such systems can exhibit parasitic instabilities. We show discretizations that lead, appropriately, to 2n first order equations, the proper degree of degeneracy. Such discretizations have been developed for canonical systems and for two non-canonical systems, the
magnetic field line equations (MFL) and the guiding center (GC) equations.

These DVI schemes have first order accuracy. We show methods of improving their accuracy. One is the development of second order accuracy. We exhibit second order schemes for canonical systems and for the noncanonical MFL and GC examples. We observe that taking a first order accurate scheme and composing it with its adjoint to obtain second order accuracy, may not work for such schemes because the scheme's conserved two form and that of its adjoint preserve different two-forms, and their composition does not appear to conserve any two-form. Also, higher order composition schemes require nonequal time substeps which can prevent these schemes as well from having a conserved two-form.

The second development is that of nonuniform and adaptive time stepping. This is achieved by writing an action in the extended phase space, leading to an extended phase space Hamiltonian. We show discretizations that lead to properly degenerate integrators (DVIs) in this extended phase space, both for canonical systems and for the field line and guiding center examples of systems with non-canonical variables.

We will show numerical examples indicating the good conservation properties of symplectic integrators, second order accuracy, and higher accuracy/efficiency due to non-uniform time stepping.

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