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In a previous paper, we developed a new particle-in-cell method for theVlasov-Maxwell system in which the electromagnetic fields and the equations ofmotion for the particles were cast in terms of scalar and vector potentialsthrough a Hamiltonian formulation. This paper extends this new class of methodsby focusing on the enforcement the Lorenz gauge condition in both exact andapproximate forms using co-located meshes. A time-consistency property of theproposed field solver for the vector potential form of Maxwell's equations isestablished, which is shown to preserve the equivalence between thesemi-discrete Lorenz gauge condition and the analogous semi-discrete continuityequation. Using this property, we present three methods to enforce asemi-discrete gauge condition. The first method introduces an update for thecontinuity equation that is consistent with the discretization of the Lorenzgauge condition. The second approach we propose enforces a semi-discretecontinuity equation using the boundary integral solution to the fieldequations. The third approach introduces a gauge correcting method that makesdirect use of the gauge condition to modify the scalar potential and uses localmaps for both the charge and current densities. The vector potential comingfrom the current density is taken to be exact, and using the Lorenz gauge, wecompute a correction to the scalar potential that makes the two potentialssatisfy the gauge condition. We demonstrate two of the proposed methods in thecontext of periodic domains. Problems defined on bounded domains, includingthose with complex geometric features remain an ongoing effort. However, thiswork shows that it is possible to design computationally efficient methods thatcan effectively enforce the Lorenz gauge condition in an non-staggered PICformulation.

A Particle-in-cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition