A Structure-Preserving Approach to Maxwell-Vlasov

Geometric integrators preserve aspects of the exact solutions to differential equations. Symplectic integrators, which have discrete flow maps that conserve a canonical 2-form, may be among the most familiar because of their use in classical mechanics; however, numerical methods that follow this general approach have been applied to a variety of equations, including those that describe fluid mechanics and electromagnetism. Suggestively, the Lagrangian which gives rise to the Maxwell-Vlasov equations contains terms that may be naturally interpreted as fluid, electromagnetic, or interaction energies. This decomposition motivates us to construct a type of splitting method for Maxwell-Vlasov, taking advantage of the structures built into its component parts. On the fluid side, we interpret the configuration space as a generalized Lie group, which falls into the framework of Euler-Poincaré reduction. On the electromagnetic side, we view the relationships among the potentials and fields in terms of the de Rham complex. With the goal of a structure-preserving numerical method for plasma dynamics in mind, we write the reduced Maxwell-Vlasov equations in a weak, variational form, then use the approximation spaces suggested by the structures of the Lagrangian's component parts to obtain a semi-discrete form of the equations. We then consider different ways to perform the time discretization and indicate potential refinements to the method.