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The eigensystem of the Euler equations

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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.