Leftover charge near the axis when strong U_theta

[haykh]

Left: SR, right: GR.

Only occurs when the particle is reflected off the axis with a large-enough $u_\theta$. Pgen to reproduce: -D pgen=debug/deposit with both engines.

[haykh]

The problem is worse when the velocity is high (ultra-relativistic). Comparison of $\nabla\cdot\boldsymbol{E}$ for ultra-relativistic velocity ($u\theta\approx 1$, top), and sub-relativistic ($u\theta\approx 10^{-2}$, bottom). Near the axis the velocity is purely tangential: $\boldsymbol{u} = \hat{\boldsymbol{\theta}} u_\theta$.

PS. A particle flying in the last cell radially without reflecting leaves no charge behind.

[haykh]

The problem also scales with the CFL number. Top: CFL = 0.5, bottom -- CFL = 0.01, for $u_\theta\approx 1.0$.

[haykh]

Solved in the latest PR #29. Two things needed to be redesigned:

• inconsistency in the $\theta$-direction filtering of currents near the axes,
• coordinate reflection of particles near the axes.

Two fixes:

• use symmetries to define the filters (e.g., $J^r{i+1/2, j, k} = (1/2) J^r{i+1/2, j, k} + (1/2) J^r_{i+1/2, j + 1, k}$, etc.)
• reflect particle velocities instead of the coordinates when the particle enters the first active cell.

Left: SR, right: GR

[haykh]

Merged #29