A continuity-preserving and divergence-cleaning algorithm based on purely and damped hyperbolic Maxwell equations in inhomogeneous media

In the numerical solution of Maxwell's equations for dynamic electromagnetic fields, the two Gauss's laws are usually not considered since they are a natural consequence of Faraday's and Maxwell-Ampère's laws if the charge conservation law is satisfied. However, when the charge conservation law is not satisfied, the numerical errors of Gauss's laws will increase unbounded, leading to numerical instability or breakdown. Unfortunately, the charge conservation law can be easily violated in self-consistent wave-particle simulations. In the meantime, the violation of Gauss's laws will also result in an increased error in the normal flux continuity. In the simulations of pure electromagnetic problems, the satisfaction of tangential field continuity across a material interface is sufficient to yield accurate numerical results. However, in a self-consistent wave-particle simulation, the normal components are as important as the tangential components, since they are critical in predicting the particle kinetics. In this paper, a divergence-cleaning method is presented to enforce Gauss's laws and normal flux continuity by introducing auxiliary variables and damping terms into Maxwell's equations in inhomogeneous media, which yield hyperbolic and mixed hyperbolic-parabolic Maxwell equations. The numerical solution schemes of the resulting purely and damped hyperbolic Maxwell equations are introduced under the framework of the discontinuous Galerkin time-domain method. It is shown through several numerical examples that the proposed method is able to preserve the continuity conditions for both tangential field components and normal flux components, and is effective in cleaning the numerical errors in both Gauss's laws.