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. $ c 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Time-step dependent force interpolation scheme for suppressing numerical Cherenkov instability in relativistic particle-in-cell simulations

“…The fully explicit Maxwell solvers in Ref. [11], which modifies the spatial derivative stencil in Faraday's equation and keeps the spatial derivative stencil in Ampere's equation, are compatible with the charge conserving deposition scheme [3]. By choosing the coefficients for the stencil in Faraday's equation, the dispersion error can be fourth order, i.e.…”

“…The dispersion relation of electromagnetic waves for the Maxwell solvers with modified spatial derivative in Faraday's equation is [11] s 2 ω = s 2 1 A 1 + s 2 2 A 2 + s 2 3 A 3 (C.1)…”

“…In a PIC simulation, if the characteristic wavelength λ or wavevector k = 2π λ satisfies k∆x = 2π∆x λ 1, then the dispersion properties near k∆x = 0 is much more important than the dispersion properties in the rest of the k space. An approach for minimizing the dispersion error [11] can be done in general cases, but we focus on reducing the dispersion error near k∆x = 0. We expand the phase velocity v g = ω/k to second order using the the spherical coordinates for the wave vectors (k 1 , k 2 , k 3 ) = (k sin θ cos φ, k sin θ sin φ, k cos θ)…”

“…The WT scheme recovers the momentum conserving scheme [9,10] for the case where the time step is ∆t = ∆x1 2c and the grid spacings in all directions are the same. In the WT scheme, the lowest order NCI growth rate still vanishes if the spatial derivative stencil in the Faraday equation are modified [11] while keeping the spatial derivative stencil in the Ampere equation. However, the asymptotic expression for NCI growth rate only shows the lowest order growth rate.…”

Efficient retrieval of phase information from real-valued electromagnetic field data

“…However, the discretization of a simulation may cause a different dispersion relation ω(k) = ω grid ( #‰ k ) [3,[6][7][8][9]] to be at work. In order to produce an accurate result from the simulation data, it might be necessary to insert the grid dispersion relation into our ansatz instead of ω vac.…”