<p> Geometric integrators yield high-fidelity numerical results by retaining conservation laws in the time advance. A particularly powerful class of geometric integrators is symplectic integrators, which are widely used in orbital mechanics and accelerator physics. An important application presently lacking symplectic integrators is the guiding center motion of magnetized particles represented by non-canonical coordinates. Because guiding center trajectories are foundational to many simulations of magnetically confined plasmas, geometric guiding center algorithms have high potential for impact. The motivation is compounded by the need to simulate long-pulse fusion devices, including ITER, and opportunities in high performance computing, including the use of petascale resources and beyond. </p><p> This dissertation uses a systematic procedure for constructing geometric integrators &mdash; known as variational integration &mdash; to deliver new algorithms for guiding center trajectories and other plasma-relevant dynamical systems. These variational integrators are non-trivial because the Lagrangians of interest are degenerate - the Euler-Lagrange equations are first-order differential equations and the Legendre transform is not invertible. The first contribution of this dissertation is that variational integrators for degenerate Lagrangian systems are typically <i>multistep methods.</i> Multistep methods admit parasitic mode instabilities that can ruin the numerical results. These instabilities motivate the second major contribution: degenerate variational integrators. By replicating the degeneracy of the continuous system, degenerate variational integrators avoid parasitic mode instabilities. The new methods are therefore robust geometric integrators for degenerate Lagrangian systems. </p><p> These developments in variational integration theory culminate in one-step degenerate variational integrators for non-canonical magnetic field line flow and guiding center dynamics. The guiding center integrator assumes coordinates such that one component of the magnetic field is zero; it is shown how to construct such coordinates for nested magnetic surface configurations. Additionally, collisional drag effects are incorporated in the variational guiding center algorithm for the first time, allowing simulation of energetic particle thermalization. Advantages relative to existing canonical-symplectic and non-geometric algorithms are numerically demonstrated. All algorithms have been implemented as part of a modern, parallel, ODE-solving library, suitable for use in high-performance simulations.</p>

A statistical equilibrium theory is developed which characterizes the large-scale coherent structures that emerge during the course of the evolution of an ideal two-dimensional magnetofluid. Macrostates are defined to be local joint probability distributions, or Young measures, on the values of the fluctuating magnetic field and velocity field at each point in the spatial domain. The most probable macrostate is found by maximizing a Kullback-Liebler entropy functional subject to constraints dictated by the conserved integrals of the ideal dynamics. This maximum entropy macrostate is, for each point in the spatial domain, a Gaussian probability distribution, whose local mean is an exact stationary solution of the evolution equations of the magnetohydrodynamic system. The predictions of the statistical equilibrium model are found to be in excellent qualitative and quantitative agreement with recent high resolution numerical simulations of turbulence in slightly dissipative two-dimensional magnetofluids.

We consider strong wave propagation in the generalized compressible Euler equations. Our results include pairwise interactions of nonlinear waves, smooth wave propagation, formation of gradient blowup and several exact examples. In particular, we directly generalize P.Lax’s gradient blowup results for conservation laws with two variables to the generalized compressible Euler equations.