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 .. math:: \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} along with the divergence relations .. math:: \nabla\cdot\mathbf{E} &= \frac{\varrho_c}{\epsilon_0} \\ \nabla\cdot\mathbf{B} &= 0. Here, :math:`\mathbf{E}` is the electric field, :math:`\mathbf{B}` is the magnetic flux density, :math:`\epsilon_0`, :math:`\mu_0` are permittivity and permeability of free space, and :math:`\mathbf{J}` and :math:`\varrho_c` are specified currents and charges respectively. The speed of light is determined from :math:`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, .. math:: \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. The eigenvalues of this system are :math:`\{-c,-c,c,c,0,0\}`. The right eigenvectors of the flux Jacobian are given by the columns of the matrix .. math:: 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]. The left eigenvectors are the rows of the matrix .. math:: 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]. 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 .. math:: \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. Here, :math:`\psi` and :math:`\psi` are correction potentials for the electric and magnetic field respectively and :math:`\chi` and :math:`\gamma` are dimensionless factors that control the speed at which the errors are propagated. In 1D these equations can be written as, ignoring sources, .. math:: \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. The eigenvalues of this system are :math:`\{-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 .. math:: 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]. The left eigenvectors are the rows of the matrix .. math:: 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]. .. [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.