# Electrodynamic Field Solver

We use a three dimensional hyperbolic solver for the conversation law. In order to capture electrodynamic phenomena, Maxwell’s equations have to be solved. A hyperbolic divergence cleaning approach is applied to prevent divergence errors from accumulating. The purely hyperbolic Maxwell’s equations (PHM) in the time-domain are

$

\cfrac{\partial \vec{E}}{\partial t} = c^2 \nabla \times \vec{B} - \chi c^2 \nabla \varPsi - \cfrac{\vec{j}}{\varepsilon_0} , \label{equ:PHM8.1}\\
\cfrac{\partial \vec{B}}{\partial t} = - \nabla \times \vec{E} - \chi \nabla \varPhi , \\
\cfrac{\partial\, \varPhi}{ \partial t} = - \chi c^2 \nabla \cdot \vec{B} , \\
\cfrac{\partial\, \varPsi}{\partial t} = - \chi \nabla \cdot \vec{E} + \chi \cfrac{\rho}{\varepsilon_0}.
$

A test case of the sole PHM solver is a oscillating dipole.