Preserving the divergence constraint in time domain simulations of the Vlasov-Maxwell equations

In the continuous setting, if the sources of Maxwell's equation satisfy the continuity equation and satisfy the divergence constraints at initial time, they satisfy those constraints for all time. This divergence constraint is in particular satisfied by sources computed from the Vlasov equation. However this is no more true at the discrete level.

Different methods have been developed to enable stable long time simulations of the Vlasov-Maxwell equations. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equation, 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.

The source correction method can be applied quite naturally for conforming Finite Elements in the Framework of Finite Element Exterior Calculus. However, one needs to be careful when projecting the sources. A more difficult question is how this can be extended to Discontinuous Galerkin formulations. We shall see how the Finite Element Exterior Calculus can also be used in this setting along with appropriate projections and definition of the discrete divergence to get long time stability in this case.