Title: A Systematic Approach to Numerical Dispersion in Maxwell Solvers,
Abstract: The finite-difference time-domain (FDTD) method is a well established method
for solving the time evolution of Maxwell's equations. Unfortunately the scheme
introduces numerical dispersion and therefore phase and group velocities which
deviate from the correct values. The solution to Maxwell's equations in more
than one dimension results in non-physical predictions such as numerical
dispersion or numerical Cherenkov radiation emitted by a relativistic electron
beam propagating in vacuum.
Improved solvers, which keep the staggered Yee-type grid for electric and
magnetic fields, generally modify the spatial derivative operator in the
Maxwell-Faraday equation by increasing the computational stencil. These
modified solvers can be characterized by different sets of coefficients,
leading to different dispersion properties. In this work we introduce a norm
function to rewrite the choice of coefficients into a minimization problem. We
solve this problem numerically and show that the minimization procedure leads
to phase and group velocities that are considerably closer to $c$ as compared
to schemes with manually set coefficients available in the literature.
Depending on a specific problem at hand (e.g. electron beam propagation in
plasma, high-order harmonic generation from plasma surfaces, etc), the norm
function can be chosen accordingly, for example, to minimize the numerical
dispersion in a certain given propagation direction. Particle-in-cell
simulations of an electron beam propagating in vacuum using our solver are
provided.