Dr Jonathan Gratus
PIC codes are used extensively in both conventional beam dynamics and laser-plasma interactions. Macro-particles are essential given the large number of particles one wishes to simulate. To calculate the input for Maxwell's equations, the charge and current of each macro-particle is \smeared" or \diluted" over the nearest cell points.
We propose an improved method of modelling the macro-particles. Current PIC codes only record the position and velocity of each macro-particle. By contrast, our new method will record higher moments. These new super-macro-particles (SMPs) will then have to be smeared over a large number of cells, the higher moments being used to reconstruct the phase space distribution of charge. This method will increase
the sophistication of the algorithm, albeit at a small cost compared to the usual method of simply increasing the number of macro-particles. The potential advantages, however, may be huge, equivalent to replacing tens or hundreds of macro-particles with a handful of these new SMPs. Indeed, if enough moments are chosen, an entire bunch of particles could be represented by a single SMP.
There are two key steps for the successful implementation of this algorithm. The rest is to calculate the correct dynamical equations for the moments. The second is the reconstruction of the phase space distribution for each SMP. Using powerful techniques of differential geometry, JG has already shown that the equations of motion for the quadrupole moment have unexpected features. 
By October 2019 a general analytic formula for both the dynamics of the SMPs and the reconstruction of the distribution will have been calculated and we are looking for a PhD student to help implement this algorithm. As a prelude to full-scale 3-dimensional implementation, lower-dimensional codes could be run and benchmarked against existing codes.
The SMP approach will be excellent at calculating CSR wakes where the distance between the centres is large. There is also the possibility of incorporating space-charge into SMPs. Radiation reaction is difficult to model for standard macro-particles as the only information is position an velocity. However, since SMP have information about higher moments one can model space-charge by altering the equation for the second and
Currently, we wish to develop the algorithm and demonstrate proof of principle. However, it is hoped the code could be applied to several existing projects, in particular, FELs where the CSR is so important.
As far as is known to the proposers (after searching) no similar approach has been considered. Although several people, starting with Esirkepov  and including Vay et al  do consider smearing the macro-particles over several cells with a chosen shape.
Program of PhD: We will search for a student with an interest in programming. In the rest year, as well as the CI lectures, the student will learn the necessary differential geometry and theory of distributions to understand the nature of the algorithm. They will also improve their coding skills maybe leading to writing a simple 1d PIC code. She or he will then implement a 1d SMP can compare the result. In years two and three the student will familiarise themselves with an existing 3d PIC code and implement the SMP.
 Gratus, J., Banaszek, T. \The correct and unusual coordinate transformation rules for electromagnetic quadrupoles" Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. Volume 474, 2213 (2018)
 \T.Zh. Esirkepov, Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor", Computer Physics Communications, Volume 135, Issue 2, 144 (2001)
 J.-L. Vay, C.G.R. Geddes, E. Cormier-Michel, D.P. Grote, \Numerical methods for instability mitigation in the modelling of laser wakeeld accelerators in a Lorentz- boosted frame," Journal of Computational Physics, Volume 230, Issue 15, 5908 (2011)