Improved accuracy in degenerate variational integrators for guiding centre and magnetic field line flow

• Published in: Journal of plasma physics Vol. 88; no. 2
• Main Authors: Burby, Joshua W., Finn, John M., Ellison, C. Leland
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.04.2022; Cambridge University Press (CUP)
• Subjects: Accuracy; Approximation; Dynamical systems; Hamiltonian functions; Hamiltonian Methods in Plasma Physics; Integrators; Magnetic fields; Plasma physics; Plasmas (physics); plasma dynamics
• ISSN: 0022-3778; 1469-7807
• DOI: 10.1017/S0022377821001136

First-order-accurate degenerate variational integration (DVI) was introduced in Ellison et al. (Phys. Plasmas, vol. 25, 2018, 052502) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introduce second-order-accurate DVI schemes, both with and without non-uniform time stepping. We show that it is not in general possible to construct a second-order scheme with a preserved two-form by composing a first-order scheme with its adjoint, and discuss the conditions under which such a composition is possible. We build two classes of second-order-accurate DVI schemes. We test these second-order schemes numerically on two systems having non-canonical variables, namely the magnetic field line and guiding centre systems. Variational integration for Hamiltonian systems with non-uniform time steps, in terms of an extended phase space Hamiltonian, is generalized to non-canonical variables. It is shown that preservation of proper degeneracy leads to single-step (one-step) methods without parasitic modes, i.e. to non-uniform time step DVIs. This extension applies to second-order-accurate as well as first-order schemes, and can be applied to adapt the time stepping to an error estimate.