Lagrangian And Symmetry Structure Of The Divergence Cleaning Model Based On Generalized Lagrange Multipliers

A field theoretical method for the treatment of the often violated charge conservation laws in computational electrodynamics is presented. This approach explores the basic symmetry features of Maxwell theory and the analogy between the gauge field anomalies of quantum field theory and the violation of charge conservation law on the lattice, in Lorentz covariant Lagrangian formalism. A mathematical construction of the counter terms to the anomalous charge conservation law is proposed, and thereby a consistent theory for the Generalized Lagrange Multiplier$\mathsf{(GLM)}$method is presented, which has so far lacked a concrete theoretical framework. Based on the established theoretical framework, the$\mathsf{GLM}$method has been further extended and the question regarding whether$\mathsf{GLM}$method solves "right" equations is answered. This extended$\mathsf{GLM}$method with new insight is then applied to magnetohydrodynamics$\mathsf{(MHD)}$and recently proposed "shallow water"$\mathsf{MHD}$. In particular, a$\mathsf{GLM}$-based Godunov-type finite-volume solver for the "shallow water"$\mathsf{MHD}$system on the Cartesian grid has been developed, and the introduced theoretical framework for$\mathsf{GLM}$model is validated. In addition, associated new analytic features of the "shallow water"$\mathsf{MHD}$system is also presented.