Vlasov iteration

The equations of beamtracer.solver.vlasov, declared on the code that implements them and checked by the tests.

`apply_compensation`

functionsource

Return rho with the compensation of every region applied.

compensation: $\rho_b \leftarrow (1 - f)\, \rho_b \quad \text{on the node columns } x_\mathrm{min} \le x_i \le x_\mathrm{max}$

Assumes

$f \in [0, 1]$ the compensated share, e.g. by the electrons of the background gas; regions applied in turn, so overlapping ones multiply

functionsource

Run iterations Vlasov iterations.

iteration 0: $\phi^0 = \text{the vacuum solution, with the free nodes at } x < x_\mathrm{plasma} \text{ held at } U_p \text{ (with a plasma)}$
solve: $\nabla^2 \phi^{n} = -\frac{\rho_b^{n} + \rho_p(\phi^{n})}{\varepsilon_0}, \qquad \text{Newton from } \phi^{n-1}$
relaxation: $\rho_b^{n+1} = (1 - \omega)\, \rho_b^{n} + \omega\, \rho_\mathrm{new}(\phi^{n}), \qquad \rho_b^1 = \rho_\mathrm{new}(\phi^0)$
convergence: $\max |\phi^{n} - \phi^{n-1}| < \text{tolerance}$

Assumes

$\rho_\mathrm{new}$ the compensated deposit of the beam traced in $\phi^n$ , re-seeded identically every iteration; $\omega$ = relaxation in $(0, 1]$ acts on the charge density, not on the potential
the Boltzmann species built once, at $n = 1$ , with $\rho_0$ given or the beams' $\sum J / v_0$
at most iterations; an unconverged iteration returns its last state without a warning (stats['converged'] is 0); the returned trace is that of the newest potential
with MPI every rank solves the whole field and the deposits are summed

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