Abstract. In this article we study a charge-conserving finite-element particle scheme for the Maxwell-Vlasov system that is based on a div-conforming representation of the electric field and we propose a high-order deposition algorithm for smooth particles with piecewise polynomial shape. The numerical performances of the method are assessed with an academic beam test-case, and it is shown that for an appropriate choice of the particle parameters the efficiency of the resulting method overcomes that of similar finite-element schemes using point particles.Résumé. Cet article présente un schémaéléments-finis conservant la charge pour le système de Vlasov-Maxwell basé sur une représentation div-conforme du champélectrique, ainsi qu'un algorithme d'ordreélevé pour déposer le courant porté par des particules régulières polynômiales par morceaux. Les performances numériques du schéma couplé sontévaluéesà l'aide d'un cas test académique de faisceau, et nous montrons que pour un choix approprié des paramètres des particules, cette méthode se révèle plus efficace que d'autres schémas similaires couplés avec des particules ponctuelles.

The aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the Vlasov-Maxwell equations and the Maxwell equations with sources. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equations, for which additional boundary conditions are in some cases non trivial to find. The source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete Gauss' law that needs to be defined in accordance with the discretization of the Maxwell propagation operator.