# The eigensystem of the Maxwell equations with extension to perfectly hyperbolic Maxwell equations¶

## Eigensystem of Maxwell equations¶

In this document I list the eigensystem of the Maxwell equations. Maxwell’s equations consist of the curl equations

$\begin{split}\frac{\partial \mathbf{B}}{\partial t} + \nabla\times\mathbf{E} &= 0 \\ \epsilon_0\mu_0\frac{\partial \mathbf{E}}{\partial t} - \nabla\times\mathbf{B} &= -\mu_0\mathbf{J}\end{split}$

along with the divergence relations

$\begin{split}\nabla\cdot\mathbf{E} &= \frac{\varrho_c}{\epsilon_0} \\ \nabla\cdot\mathbf{B} &= 0.\end{split}$

Here, $$\mathbf{E}$$ is the electric field, $$\mathbf{B}$$ is the magnetic flux density, $$\epsilon_0$$, $$\mu_0$$ are permittivity and permeability of free space, and $$\mathbf{J}$$ and $$\varrho_c$$ are specified currents and charges respectively. The speed of light is determined from $$c=1/(\mu_0\epsilon_0)^{1/2}$$.

These are linear equations and hence the eigensytem is independent of the value of the electromagnetic fields. In 1D Maxwell equations can be written as, ignoring sources,

$\begin{split}\frac{\partial }{\partial t} \left[ \begin{matrix} E_x \\ E_y \\ E_z \\ B_x \\ B_y \\ B_z \end{matrix} \right] + \frac{\partial }{\partial x} \left[ \begin{matrix} 0 \\ c^2B_z \\ -c^2B_y \\ 0 \\ -E_z \\ E_y \end{matrix} \right] = 0.\end{split}$

The eigenvalues of this system are $$\{-c,-c,c,c,0,0\}$$. The right eigenvectors of the flux Jacobian are given by the columns of the matrix

$\begin{split}R = \left[ \begin{matrix} 0&0&0&0&1&0 \\ 1&0&1&0&0&0 \\ 0&1&0&1&0&0 \\ 0&0&0&0&0&1 \\ 0&{{1}\over{c}}&0&-{{1}\over{c}}&0&0 \\ -{{1}\over{c}}&0&{{1}\over{c}}&0&0&0 \end{matrix} \right].\end{split}$

The left eigenvectors are the rows of the matrix

$\begin{split}L = \left[ \begin{matrix} 0&{{1}\over{2}}&0&0&0&-{{c}\over{2}} \\ 0&0&{{1}\over{2}}&0&{{c}\over{2}}&0 \\ 0&{{1}\over{2}}&0&0&0&{{c}\over{2}} \\ 0&0&{{1}\over{2}}&0&-{{c}\over{2}}&0 \\ 1&0&0&0&0&0 \\ 0&0&0&1&0&0 \end{matrix} \right].\end{split}$

## Eigensystem of Perfectly Hyperbolic Maxwell equations¶

The perfectly hyperbolic Maxwell equations are a modification of the Maxwell equations that take into account the divergence relations. The modified equations explicitly “clean” divergence errors and are a hyperbolic generalization of the Hodge project method commonly used in electromagnetism to correct for charge conservation errors. See [munz_2000], [munz_2000b], [munz_2000c] for details.

These equations are written as

$\begin{split}\frac{\partial \mathbf{B}}{\partial t} + \nabla\times\mathbf{E} + \gamma \nabla\psi &= 0 \\ \epsilon_0\mu_0\frac{\partial \mathbf{E}}{\partial t} - \nabla\times\mathbf{B} + \chi \nabla \phi &= -\mu_0\mathbf{J} \\ \frac{1}{\chi}\frac{\partial \phi}{\partial t} + \nabla\cdot\mathbf{E} &= \frac{\varrho_c}{\epsilon_0} \\ \frac{\epsilon_0\mu_0}{\gamma}\frac{\partial \psi}{\partial t} + \nabla\cdot\mathbf{B} &= 0.\end{split}$

Here, $$\psi$$ and $$\psi$$ are correction potentials for the electric and magnetic field respectively and $$\chi$$ and $$\gamma$$ are dimensionless factors that control the speed at which the errors are propagated.

In 1D these equations can be written as, ignoring sources,

$\begin{split}\frac{\partial }{\partial t} \left[ \begin{matrix} E_x \\ E_y \\ E_z \\ B_x \\ B_y \\ B_z \\ \phi \\ \psi \end{matrix} \right] + \frac{\partial}{\partial x} \left[ \begin{matrix} \chi c^2 \phi \\ c^2B_z \\ -c^2B_y \\ \gamma \psi \\ -E_z \\ E_y \\ \chi E_x \\ \gamma c^2B_x \end{matrix} \right] = 0.\end{split}$

The eigenvalues of this system are $$\{-c\gamma, c\gamma, -c\chi, c\chi, -c, -c, c, c\}$$. The right eigenvectors of the flux Jacobian are given by the columns of the matrix

$\begin{split}R = \left[ \begin{matrix} 0&0&1&1&0&0&0&0 \\ 0&0&0&0&1&0&1&0 \\ 0&0&0&0&0&1&0&1 \\ 1&1&0&0&0&0&0&0 \\ 0&0&0&0&0&{{1}\over{c}}&0&-{{1}\over{c}} \\ 0&0&0&0&-{{1}\over{c}}&0&{{1}\over{c}}&0 \\ 0&0&-{{1}\over{c}}&{{1}\over{c}}&0&0&0&0 \\ -c&c&0&0&0&0&0&0 \end{matrix} \right].\end{split}$

The left eigenvectors are the rows of the matrix

$\begin{split}L = \left[ \begin{matrix} 0&0&0&{{1}\over{2}}&0&0&0&-{{1}\over{2\,c}} \\ 0&0&0&{{1}\over{2}}&0&0&0&{{1}\over{2\,c}} \\ {{1}\over{2}}&0&0&0&0&0&-{{c}\over{2}}&0 \\ {{1}\over{2}}&0&0&0&0&0&{{c}\over{2}}&0 \\ 0&{{1}\over{2}}&0&0&0&-{{c}\over{2}}&0&0 \\ 0&0&{{1}\over{2}}&0&{{c}\over{2}}&0&0&0 \\ 0&{{1}\over{2}}&0&0&0&{{c}\over{2}}&0&0 \\ 0&0&{{1}\over{2}}&0&-{{c}\over{2}}&0&0&0 \end{matrix} \right].\end{split}$
 [munz_2000] C.-D Munz, P. Omnes, R. Schneider and E. Sonnendruer and U. Voss, “Divergence Correction Techniques for Maxwell Solvers Based n a Hyperbolic Model”, Journal of Computational Physics, 161, 484-511, 2000.
 [munz_2000b] C.-D Munz, P. Omnes, and R. Schneider, “A three-dimensional finite-volume solver for the Maxwell equations with divergence cleaning on unstructured meshes”, Computer Physics Communications, 130, 83-117, 2000.
 [munz_2000c] C.-D Munz and U. Voss, “A Finite-Volume Method for the Maxwell Equations in the Time Domain”, SIAM Journal of Scientific Computing, 22, 449-475, 2000.