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.