Dissertation Title: On Hamiltonian structure preserving discretizations of plasma and nonlinear wave models

Abstract: Every physical model has a geometric interpretation. In classical physics, this geometric framework lies in its Lagrangian and Hamiltonian formulation. Even dissipative dynamics can be formulated in terms of gradient flows. While traditional numerical treatments of these models largely neglect this underlying structure, the desire to exactly enforce conservation laws at the discrete level encourages one to consider geometric structure in the design of numerical methods for evolution equations. This work studies the preservation of Hamiltonian structure in the discretization of plasma and nonlinear optical models. A general Hamiltonian structure preserving discretization for nonlinear Hodge wave equations is derived using finite element exterior calculus. This general framework is then specialized to derive an energy and Gauss conserving time-domain solver for Maxwell's equations in nonlinear media. This method is applied to study a model of the ponderomotive force and cubicly nonlinear electromagnetic media. Finally, two varieties of particle-based Hamiltonian reductions of the Vlasov equation are derived which assist in the design of structure preserving particle-in-cell (PIC) methods. In particular, the analysis provides criteria for structure-preserving grid-interpolation and filtering in PIC methods, both being components of a PIC algorithm which might undermine Hamiltonian structure preservation if implemented naïvely.