Physical Symmetries and Hyperbolic GLM Divergence Correction Scheme for Maxwell and MHD Equations

The charge conservation laws in general are not strictly obeyed in computational electromagnetics and MHD, due to the presence of various types of numerical errors. The violation of the charge conservation laws in field theoretical viewpoint is a consequence of the broken gauge symmetry in the computational space. In this paper, we present a new field theoretical method for the treatment of the often violated charge conservation laws in computational electrodynamics and MHD, which is consistent with the symmetries of Maxwell theory, namely the Lorentz-, gauge- and duality symmetries. This method under particular constraints reduces to the existing GLM scheme which has so far lacked a concrete theoretical framework, and thereby provides a new insight into the GLM scheme. The central idea of our divergence correction scheme is the implementation of the physically consistent counter term Ansätze to Maxwell and MHD equations, for the restora- tion of the gauge symmetry. One of the main advantages of our method is that the divergence conditions for the charge conservations are implemented into the Maxwell and MHD equations in a hyperbolic form, rather than the genuine elliptic form, and it can be easily implemented into the existing codes for Maxwell and MHD solvers via operator splitting Ansatz.