Gradient-based longitudinal space charge calculation

[austin-hoover]

This PR adds the option to calculate the longitudinal space charge kick in LSpaceChargeCalc using the gradient of the charge density. The current solver calculates the kick in frequency space using the impedance formulation.

An option has also been added to set the number of FFT modes used to calculate the energy kicks in the impedance-based formulation. Before it was set to half the number of grid cells.

[austin-hoover]

It looks good in some initial tests; will need to think about how to benchmark.

[shishlo]

So, what about the benchmark? Was it successful? As I understand frequency approach allows to cut or to keep high frequency noise. What about the suggested new gradient method? Could you reproduce the results of the paper:
https://journals.aps.org/prab/abstract/10.1103/PhysRevSTAB.7.094201
So, bottom line: would it better to keep the old frequency based method and add a new one with derivatives?

[austin-hoover]

This PR doesn't replace the frequency approach, just adds option to use gradient instead.

One option for a benchmark would be a uniform disk in x-y plane within conducting pipe of radius b, and then a Gaussian z distribution. The energy kick from the 1D and 2D-sliced solvers should agree with the analytic result.

I've tested this 1D solver against 2D-sliced for bunch compression simulations in the SNS ring, and they give similar results. But we do expect differences in that case since the transverse distribution is not uniform.

I'm not sure about the space-charge-sustained microbunching. We should definitely try to reproduce these simulations, but it might be better to start with a simpler benchmark.

[shishlo]

I know that people are using this node in J-PARC simulations. What method will be used by default? Also, could you add an example of how to use and comparison of two methods?

[austin-hoover]

We will keep frequency-based model as default. Eventually documentation should explain the difference between the methods.

[austin-hoover]

Added benchmark for Gaussian distribution. the initial distribution has zero energy spread and a Gaussian longitudinal distribution. We assume the transverse distribution is uniform within radius $a$ and that there is a conducting pipe at radius $b > a$.

Frequency-based: number of modes

Used 10,000 particles and 32 bins. Scanned number of FFT mode: {2, 4, 8, 16}. Analytic result is shown in red.

Gradient-based: number of particles

Used 32 bins. Scanned number of macroparticles: {1e4, 1e5, 1e6}.

Gradient-based: smoothing

Used 100,000 particles and 32 bins. Scanned gradient smoothing {off, on}.

[shishlo]

Thanks, Austin! To my mind, that makes sense. I’m in favor of the merger.