Maxwell-Vlasov equations for laboratory plasmas: Conservation laws and approximation schemes

Laser plasma and beam plasma acceleration are promising routes towards a new generation of particle accelerators and the numerical solutions of Maxwell-Vlasov equations are used to design and interpret the experiments. We review the variational formulation of the Maxwell-Vlasov equations and derive the balance equations for energy, momentum and charge in local form in the fully kinetic framework. This is preparatory to a work in progress aimed at investigating the conservation properties directly from the action principle in discretized form. The Particle-in-Cell (PIC) numerical schemes are reviewed with a particular focus on the local balance equations leading to global conservation laws. We provide arguments to show that momentum and energy conservation are both satisfied or violated, depending on the numerical charge conservation, which remains the major problem in the PIC framework. In schemes where the charge is locally not conserved, the Poisson's equation has to be enforced at any time to avoid numerical instabilities and anomalous heating. We propose a general scheme to enforce the Poisson's equation in terms of electromagnetic potentials under the relativistic invariant Lorentz gauge. This scheme is based on solving the wave equations for the electromagnetic potentials, with a modest additional computational load, and can be applied to both the case of beam driven and the laser driven plasma acceleration one.