The electromagnetic Darwin model for intense charged particle beams

The theoretical and numerical properties of the electromagnetic Darwin model for intense charged particle beams are investigated. The model neglects the transverse displacement current in Ampereʹs law and results in the elimination of high-frequency transverse electromagnetic waves and the associated retardation effects in the Vlasov–Maxwell equations. In this paper, two numerical schemes are presented for the purpose of circumventing the numerical instabilities associated with the presence of E T [ ≡ - ( 1 / c ) ∂ A / ∂ t ] in the equations of motion for particle codes, where A is the vector potential. The first relies on higher-order velocity moments for closure, and the other replaces the mechanical momentum, p = γ m v , by the canonical momentum, P = p + ( q / c ) A , as the phase-space variable. The properties of these simulations schemes in the laboratory frame as well as in the beam frame are also discussed. These new numerical methods are most suitable for studying Weibel and two-stream instabilities in heavy ion fusion research.