# Parallel Spectral Numerical Methods/Maxwell's Equations

Maxwell’s equations are a set of hyperbolic partial differential equations that describe the propagation of electric and magnetic fields. They can be numerically approximated by several different methods, common methods include: finite element, finite differences, and spectral methods. In this section, we wish to explain how to develop a spectral method scheme to solve Maxwell's equations. To do so, we first introduce Maxwell's equations.

Maxwell’s equations for an observer in the laboratory frame, where no free charges and currents are present, can be written as:

$\vec{D}_t - \nabla \times \vec{H} = 0$

( 1)

$\vec{B}_t + \nabla \times \vec{E} = 0$

( 2)

with the relation between the electromagnetic components given by the constitutive relations:

$\vec{D} = \vec{\zeta}_e(\bar \varepsilon, \vec{E}) = \varepsilon_o \vec{E} + \varepsilon_o\vec{\Phi}_e(\bar \varepsilon, \vec{E}) + \vec{\upsilon}\times\vec{H}$

( 3)

$\vec{B} = \vec{\zeta}_h(\bar \mu, \vec{H}) = \mu_o \vec{H} + \mu_o\vec{\Phi}_h(\bar \mu, \vec{H}) + \vec{\upsilon}\times\vec{E}$

( 4)

where $\bar \varepsilon$ and $\bar \mu$ are the electromagnetic material properties, analytically they are symmetrical second rank tensors with non zero off-diagonal entries; $\varepsilon_o$ and $\mu_o$ are the vacuum’s permittivity and permeability, respectively; the functions $\vec{\Phi}_e$ and $\vec{\Phi}_h$ are the polarization and magnetization, respectively, material’s response to the incident field; and $\vec{\upsilon}$ is known as the dichroic response of the material .[1].

1. In the context of general relativity, a similar term to the dichroic comportment will arise when transforming the fields to the co-moving field, which can be understood as the Minkowski addition to the constitutive relations