Complex-Frequency Shifted PMLs for Maxwell’s Equations With Hyperbolic Divergence Cleaning and Their Application in Particle-in-Cell Codes

Benchmarking a high-order particle-in-cell code for the simulation of a gyrotron traveling-wave tube

“…The corrugations are 15d long and the transition from the corrugation to two circular waveguides consists of two linear ramps of length 2d. Additionally, perfectly matched layers (PML) [2] are added at the output with a length of d.…”

“…Consequently, a high-order DG scheme has been applied within a particle-in-cell solver [1], replacing the standard Yee scheme. This development has been continued, leading to a three-dimensional full-wave particle-in-cell solver named PICLas, which has been benchmarked against a 30 GHz gyrotron operating in a TE 2,3 mode [2]. In contrast to finite difference methods, which use equidistant grids and a second order boundary approximation, PICLas uses body-fitted, curvilinear unstructured hexahedral meshes of arbitrary polynomial order.…”

Absorption of charged particles in Perfectly-Matched-Layers by optimal damping of the deposited current

“…However, in the original formulation of the PMLs, the presence of charged particles was not considered. In practice, a number of numerical issues can occur when charged particles reach the PML region, which can then undesirably affect the simulation at large [4,5]. While there is a large body of work on PMLs in general, the issue of charged particles reaching the PML has attracted relatively little attention.…”

“…Historically, solutions to this issue have been proposed and studied in the context of microwave devices, whereby an electron beam passes through the simulation domain, and interacts with electromagnetic modes in a cavity or a waveguide [4][5][6][7]. For instance, in [4], the authors observed that spurious electrostatic fields can build up at the boundary of the simulation domain, if charged particles are simply removed from the simulation as soon as they enter the PML region.…”

“…This does prevent the build up of spurious fields, but is only effective over timescales longer than the characteristic diffusion time, which is limited by stability constraints [4]. Alternatively, instead of removing the particles, some algorithms [5,6] allow them to propagate into the PML region, and to use the corresponding current density as a source term in the PML equations. In the case of [5], this is also combined with a propagative divergence-cleaning algorithm [9][10][11][12].…”