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

TL;DR

This paper develops second-order accurate degenerate variational integrators (DVIs) for systems with degenerate Lagrangians, improving accuracy and enabling non-uniform time stepping for guiding center and magnetic field line flow simulations.

Contribution

It introduces new second-order DVI schemes, analyzes their properties, and extends variational integration to noncanonical variables with non-uniform time steps.

Findings

Second-order DVI schemes outperform first-order in accuracy.

Preservation of degeneracy prevents parasitic modes in non-uniform time stepping.

Extension to noncanonical variables enables adaptive time stepping in complex systems.

Abstract

First-order accurate degenerate variational integration (DVI) was introduced in C. L. Ellison et. al, Phys. Plasmas 25, 052502 (2018) for systems with a degenerate Lagrangian, i.e. one in which the velocity-space Hessian is singular. In this paper we introducing 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 noncanonical variables, namely the magnetic field line and guiding center systems. Variational integration for Hamiltonian systems with nonuniform time steps, in terms of an extended phase space Hamiltonian, is generalized to noncanonical variables. It is shown that preservation of proper degeneracy leads to single-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.