Geometric integration of the Vlasov-Maxwell system with a variational   particle-in-cell scheme
  A fully variational, unstructured, electromagnetic particle-in-cell integrator is developed for integration of the Vlasov-Maxwell equations. Using the formalism of Discrete Exterior Calculus, the field solver, interpolation scheme and particle advance algorithm are derived through minimization of a single discrete field theory action. As a consequence of ensuring that the action is invariant under discrete electromagnetic gauge transformations, the integrator exactly conserves Gauss's law. 
Jonathan Squire and Hong Qin and William M. Tang
physics.plasm-ph physics.acc-ph physics.comp-ph