support convergence when number of particles per mesh cell is bounded

[alecjohnson]

With the currently implemented algorithm, iPic3D is unable to converge to the solution of the Klimontovich equation (molecular dynamics), and in order for the iPic3D solution to converge to the solution of the Vlasov equation, the number of particles per mesh cell must be taken to infinity.

Two simple changes to the algorithm would allow for convergence to the Klimontovich equation (for a fixed number of particles as the mesh resolution goes to infinity) and to the Vlasov equation (for a fixed number of particles per mesh cell as the mesh resolution goes to infinity):

1. Use a configurable number of smoothings. In the current code, the current is smoothed M=1 time before being used in the field solve, and the electric field is smoothed N=3 times after being solved. M and N should be made user-configurable numbers.

   Without this change, particle self-force is unable to become correct if the number of particles per mesh cell is bounded.

2. For convergence to the Klimontovich equation, M should equal N and we should also allow the user to specify that the unsmoothed field be retained for subsequent field solves and that the smoothed field be used only to push particles. Currently in iPic3D the smoothing overwrites the unsmoothed field.

   Evolving the unsmoothed field allows the self-force to become the correct self-force of a finite-width Gaussian-shaped particle as the mesh is refined and the number of smoothings is increased appropriately. If the unsmoothed field is overwritten with the smoothed field, then this property is lost.

For fuller justification, see

this proposal.