A Particle-in-cell Method for Plasmas with a Generalized Momentum Formulation

In this paper we formulate a new particle-in-cell method for the Vlasov-Maxwell system. Using the Lorenz gauge condition, Maxwell's equations for the electromagnetic fields are written as a collection of scalar and vector wave equations. The use of potentials for the fields motivates the adoption of a formulation for particles that is based on a generalized Hamiltonian. A notable advantage offered by this generalized formulation is the elimination of time derivatives of the potentials that are required in the standard Newton-Lorenz treatment of particles. This allows the fields to retain the full time-accuracy guaranteed by the field solver. The resulting updates for particles require only knowledge of the fields and their spatial derivatives. A method for constructing analytical spatial derivatives is presented that exploits the underlying integral solution used in the field solver for the wave equations. The field solvers considered in this work belong to a larger class of methods which are unconditionally stable, can address geometry, and leverage an $\mathcal{O}(N)$ fast summation method for efficiency, known as the Method of Lines Transpose (MOL$^T$). A time-consistency property of the proposed field solver for the vector potential form of Maxwell's equations is established, which ensures that the semi-discrete form of the proposed method satisfies the semi-discrete Lorenz gauge condition. We demonstrate the method on several well-established benchmark problems involving plasmas. The efficacy of the proposed formulation is demonstrated through a comparison with standard methods presented in the literature, including the popular FDTD method.