Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model

Usually, non-stationary numerical calculations in electromagnetics are based on the hyperbolic evolution equations for the electric and magnetic fields and leave Gaussʹ law out of consideration because the latter is a consequence of the former and of the charge conservation equation in the continuous case. However, in the simulation of the self-consistent movement of charged particles in electromagnetic fields, it is a well-known fact that the approximation of the particle motion introduces numerical errors and that, consequently, the charge conservation equation is not satisfied on the dicrete level. Then, in order to avoid the increase of errors in Gaussʹ law, a divergence cleaning step which solves a Poisson equation for a correction potential is often added. In the present paper, a new method for incorporating Gaussʹ law into non-stationary electromagnetic simulation codes is developed, starting from a constrained formulation of the Maxwell equations. The resulting system is hyperbolic, and the divergence errors propagate with the speed of light to the boundary of the computational domain. Furthermore, the basic ideas of the numerical approximation are introduced and the extended hyperbolic system is treated numerically within the framework of high-resolution finite-volume schemes. Simulation results obtained with this new technique for pure electromagnetic wave propagation and for an electromagnetic particle-in-cell computation are presented and compared with other methods.